This rule allows us to differentiate a vast range of functions. How do you differentiate #f(x)=(sin^3x^2))^(3/2)# using the chain rule? How do you differentiate #arc cot(-4sec(1/(3x^2)) )# using the chain rule? How do you differentiate #f(x)=cot(4x-x^2) # using the chain rule? How do you find the derivative of #sqrt(x^5)#? How do you use the chain rule to differentiate #y=cos(3x)#? To illustrate this, if we were asked to differentiate the function: It states: The derivative will be equal to the derivative of the outside function with respect to the inside, times the derivative of the inside function. What is the derivative of #sqrt( x^2-1)#? What is the derivative of #sqrt(x - 1)/sqrtx#? How do you differentiate #f(x)=(4x^6-3sqrt(x/(x-5)^2)+2)^2# using the chain rule? How do you differentiate # y=sqrt(sec (x^2/pi - xpi))-sec (sqrt(x^2/pi - xpi))# using the chain rule? How do you find the derivative of #y=(2/(x-1)-x^-3)^4#? What is the derivative for #f(x)=sqrt(x^2-1)#? How do you differentiate #y=e^((lnx))^2#? How do you find the derivative of #e^ [2 tan(sqrt x)]#? How do you differentiate # ln[ (2x^3)-(3x^2)+(7) ]#? How do you differentiate #y= ln(3x^2-4) #? How do you find the derivative of #3^(x^2-2)#? So the derivative of e to the g of x is e to the g of x times g prime of x. How do you use the chain rule to differentiate #f(x) = cos(lnx)#? How do you differentiate #sqrt((x+1)/(2x-1))#? How do you differentiate given #tan^2(x)#? How do you use the chain rule to differentiate #f(x) = e^(4x+9)#? What is the derivative of # ( cos (pi*x) +1 ) / x#? If #f(x)= 2 x^2 + x # and #g(x) = 2e^x + 1 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=sin(2x^3)#? Indeed, we have So we will use the product formula to get How do you differentiate #f(x)=csc(ln(1/x^4)) # using the chain rule? How do you differentiate #f(x)=cos(7-4x) # using the chain rule? How do you find the derivative of # sin^2(x/6)#? How do you find the derivative of #y=6sin(2t) + cos(4t)#? How do you find the derivative of #y = x^(cos x)#? How do you differentiate #f(x)=x-sqrt(2^x-x^2)# using the chain rule? How do you differentiate #arcsin(csc(x^3)) )# using the chain rule? How do you differentiate #f(x)=sqrt(sin^2x^2 - cos^3x)# using the chain rule? How do you differentiate # y =sin(ln(cos x)) # using the chain rule? How do you find the derivative of #f(x)=sin (1/x^2)# using the chain rule? How do you use the chain Rule to find the derivative of #sqrt(2x^3 - 3x- 4)#? What is the derivative of #sin^2(cos3x)#? How do you differentiate # y =(5x^2-x+1)^(5/2 # using the chain rule? How do you differentiate # y =(2 x^2 − 9)^(-9) # using the chain rule? How do you differentiate # y= sqrt((3x)/(2x-3))# using the chain rule? How do you find the derivative of #e^sqrt(x)#? How do you differentiate #f(x)=1/sqrt(1-x)# using the chain rule? How do you find the derivative of #sqrt(x+1)#? How do you differentiate #f(x)=sec(1/sqrtx ) # using the chain rule? How do you find the derivative of #(arctan x)^3#? How do you differentiate #f(x)=cot(e^(x)) # using the chain rule? How do you differentiate #f(x)=sqrt(1-x^2)#? How do you use the chain rule to differentiate #y=(x+1)^(-1/2)#? How do you differentiate #y=3cot(ntheta)#? If #f(x)= tan8 x # and #g(x) = e^(-3x ) #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #y = 6 cos(x^3 + 3)#? How do you find the derivative for #(-7x^2+8)^8(3x^2+9)^10#? How do you find the derivative of #sqrt(3x)#? How do you differentiate #f(x)=-2e^(x^2cosx # using the chain rule? which represents the slope of the tangent line at the point (−1,−32). How do you differentiate #h(x) = (6x-x^3)^2#? If #f(x)= sin6x # and #g(x) = sqrt(x+3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate # f(x)= (3e^x+2)^3 # using the chain rule.? How do you use the chain rule to differentiate #ln(-cosx)#? How do you find the derivative of #w=(1+4x^3)^-2#? If #f(x) =tan^2x # and #g(x) = sqrt(5x-1 #, what is #f'(g(x)) #? What is the derivative of this function #sin(x+1)#? How do you find #(dy)/(dx)# given #cos(xy^2)=y#? In short, we would use the Chain Rule when we are asked to find the derivative of function that is a composition of two functions, or in other terms, when we are dealing with a function within a function. What is the derivative of #f(x)=1/3sec^3(2x)tan(2x)-2sec(2x)tan(2x)#? How do you differentiate #f(x)=sec(4x^5)#? How do you find the derivative of # (3+sin(x))/(3x+cos(x))#? How do you find the derivative of #sqrt(4-x^2)#? How do you use the chain rule to differentiate #f(x)=sqrt(4x^3+6x)#? How do you differentiate #f(x)=(x^2+4x+6)^5#? How do you find the second derivative of #ln(sqrtx)#? (1 point) Use the chain rule to find out where z = z²y + xy2. How do you find the derivative of #y=ln(sin(x))# ? How do you differentiate # F(x) = 3x^2 + 12#? How do you differentiate #f(x)=2^(-x^2)#? How do you differentiate #f(x)=ln(3x^2)# using the chain rule? How do you find the derivative of #y= 12((x^2-7)^(1/3))#? What are the first two derivatives of #1/ln(x)#? $$ f(x) = \blue{e^{-x^2}}\red{\sin(x^3)} $$ Step 2. To put this rule into context, let’s take a look at an example: \(h(x)=\sin(x^3)\). How do you differentiate # f(x)= sin(x-2)^3 # using the chain rule.? How do you find the derivative of #g(x)=sqrt(5-3x)#? What is the derivative of #f(x) = ((x+5)/(x^2+2))^2#? How do you find the primitive function of #f(x)=1/sqrt(3-x)# and #f(x)=(4-2x)^-2# ? How do you find the derivative of #x/ sqrt (x^2 +1)#? How do you find the first and second derivative of #y=e^(alphax)sinbetax#? How do you differentiate #f(x)=ln(x^2)# using the chain rule? How do you use the chain rule to differentiate #y=(x^3+3)^5#? How do you differentiate #f(x)=(x^3-2x+3)^(3/2)# using the chain rule? How do you use the chain rule to differentiate #y=(x+1)^6/(3x-2)^5#? How do you differentiate # f(x)= (xe^x+4)^3 # using the chain rule.? How do you differentiate #f(x)=e^tan(lnx)# using the chain rule? For example, if a composite function f (x) is defined as How do you use the chain rule to differentiate #y=root4(-3x^4-2)#? How do you find the derivative of #y=tanh(6+e^(6x))#? How do you find the derivative of # cos^7(e^x)# using the chain rule? How do you find the derivative of #f(x) = e^x + e^-x / 2 #? How do you use the chain rule to differentiate #y=(-x^4-3)^-2#? How do you differentiate #y=((x^2+1)/(x^2-1))^3#? How do you differentiate #f(x)=cot(1/sqrt(x-3)) # using the chain rule? How do you find the derivative of #y=(x+8)^5#? How do you differentiate #f(x)=ln(sine^(x^2))# using the chain rule? If #f(x) =-sqrt(2x-1) # and #g(x) = 3/x^3 #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #root3(-4x)#? How do you determine #(dy)/(dx)# given #y=tan(cosx)#? If #f(x)= cos(-2 x -1) # and #g(x) = 3x^2 -5 #, how do you differentiate #f(g(x)) # using the chain rule? How do you find the derivative of #1/sqrt (x-1)#? If #f(x)= (5x -1)^3 # and #g(x) = 3x^( 2/3 ) #, what is #f'(g(x)) #? How do you differentiate #f(x) = (2x-3) ^ -2#? How do you find dy/dt given #y= 4 sin ( sqrt (1+ sqrt (x) ))#? How do you find the derivative of #sinh^79(x)#? Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. How do you find the derivative of #f(t)=(4-t)^3#? If #f(x)= sec 4 x # and #g(x) = 2 x #, how do you differentiate #f(g(x)) # using the chain rule? Other problems however, will first require the use the chain rule and in the process of doing that we’ll need to use the product and/or quotient rule. How do you use the chain rule to differentiate #y=tan(2x^3-1)#? How do you find the derivative of #r= 2theta sqrt(sec theta)# using the chain rule? How do you differentiate #f(x)=sec(e^(sqrtx-4) ) # using the chain rule? How do you differentiate #y = (x+1) (sqrt (2x-1))#? However, we rarely use this formal approach when applying the chain rule to … The following three problems require a more formal use of the chain rule. How do you differentiate #f(x) = ln((1-x^2)^(-1/2) )?# using the chain rule? The chain rule gives us that the derivative of h is . The chain rule can be used to differentiate many functions that have a number raised to a power. How do you differentiate #f(y) = e^y /y#? How do you differentiate #f(x)=ln(6 sin^2(x^3 + 2x))# using the chain rule? The outer function in this example is 2 x. How do you differentiate #y=(e^(2x)+1)^3#? Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. Answer to 2: Differentiate y = sin 5x. If #f(x)= cos 4 x # and #g(x) = -3x #, how do you differentiate #f(g(x)) # using the chain rule? How do you differentiate #f(x)=sec(1/sqrt(3x) ) # using the chain rule? Alternatively, by letting h = f ∘ g, one can also … How do you find the derivative of #ln(x^2)#? How do you differentiate #f(x)=(ln(x-(e^(tan(x^2)))))^(3/2)# using the chain rule? If #f(x)= sqrt(x^2-1 # and #g(x) = 1/x #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #f(x)=sec^2(3x^6-6x+7)tan^2(16x^-2+61cos(x^2))#? How do you differentiate #f(x)=cose^(4x)# using the chain rule.? Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. How do you find the derivative of #y=(4x+3)^-1+(x-4)^-2#? As u = 3x − 2, du/ dx = 3, so. How do you find the derivative of #sqrt(2x-3)#? (3x-10) Here in the example you see there are two functions of x, one is 56x^2 and one is (3x-10) so you must use the product rule. How do you use the chain rule to differentiate #y=4(x^2-7x+3)^(-3/4)#? How do you differentiate #f(x)=4/(x+1)^2 # using the chain rule? How do you differentiate #y=cos((1-e^(2x))/(1+e^(2x)))#? What is the derivative of #f(x)= ln (2x^3 +1)#? What is the derivative of #sqrt(x+13) / ((x-4)(root3(2x+1))#? What is the derivative of # sin^2(x) cos (x)#? Because it's so tough I've divided up the chain rule to a bunch of sort of sub-topics and I want to deal with a bunch of special cases of the chain rule, and this one is going to be called the general power rule. Example. How do you differentiate #arcsin(sqrt(cos^2(1/x) )# using the chain rule? How do you find dy/dx given #y=ln(cos x)#? How do you differentiate #f(x)=((65e^-7x)+2)^3 # using the chain rule? How do you differentiate #f(x) = e^(-5x^2)#? This calculator … How do you find the derivative of #abs(x-2)#? How do you find the derivative for #1/sqrt(1-x^2)#? We’ve been using the standard chain rule for functions of one variable throughout the last couple of sections. How do you differentiate #f(x)=1/sqrt(e^(-x^2+x) # using the chain rule? How do you differentiate # f(x)=e^(x-(x-2)^2 # using the chain rule.? How do I find the derivative of #ln(ln(2x))#? So all we need to do is to multiply dy /du by du/ dx. How do you find the derivative of #y= sqrt((x-1)/(x+1))# ? How do you differentiate #f(x)=sin(3x+1)# using the chain rule? How do you differentiate #f(x) = 4/sqrt(tan^2(1-x) # using the chain rule? How do you find the derivative of #sqrt(x+7)#? How do you use the chain rule to differentiate #y=-5/(3x^2-4)^6#? bookmarked pages associated with this title. If g is twice differentiable function and #f(x)=xg(x^2)#, how do you find f'' in terms of g, g', and g''? How do you use the chain rule to differentiate #y=cos(6x^2)#? What is the derivative of #ln(1+(1/x))#? How do you find the derivative of #s=[(t^2+1)^3+t]^-1#? How do you differentiate #f(x)=sec(e^(x)-3x ) # using the chain rule? How do you use the chain rule to differentiate #y=(x^2+4x)^(1/2)#? How do you use the chain rule to differentiate #y=7/(x^2+x)^2#? How do you differentiate #f(x)=csc(e^(x^2-5x)) # using the chain rule? How do you find the derivative of #f(x) = -5 e^{x \cos x}# using the chain rule? How do you find the derivative of #y= root3(e^x+1)# ? How do you differentiate #f(x) = (1-sqrt(3x-1))^2 # using the chain rule? How do you differentiate # f(x)=tan(e^((lnx-2)^2 ))# using the chain rule.? Derivative of the sine and cosine functions. How do you use the chain rule to differentiate #y=(x^2+3x)^(-1/2)#? Before using the chain rule, let's multiply this out and then take the derivative. If #f(x)= cos5 x # and #g(x) = e^(3+4x ) #, how do you differentiate #f(g(x)) # using the chain rule? What is the derivative of #y= (5x)/sqrt (x^2+9)#? How do you differentiate #f(x) = (3x-2)^4# using the chain rule? How do you use the chain rule to differentiate #y=sqrt(1/(x+1))#? How do you find the derivative of #sqrt(e^x)#? How do you use the chain rule to differentiate #y=(cosx/(1+sinx))^5#? How do you find the derivative of #y=cos 3x#? How do you differentiate #y=1/(x^2+1)^4#? How do you differentiate # y=sec (3 - 8x)# using the chain rule? How do you differentiate # y =( ln(3x + 2))^2# using the chain rule? How do you differentiate #y=-3sqrt(7t^3-1)#? How do you find the derivative of #y = 9tan^-1(x − (sqrt (1 + x^2))#? How do you find the derivative of #(cos x)^2 - cos x#? How do you find the derivative of #y = e^cosh(2x)#? How do you differentiate #y=sqrt( (x-1) (x-2) (x-3))#? How do you use the chain rule to differentiate #y=2(x^4+x)^-1#? How do you differentiate #5cos (2x)+ tan(3x+4)#? How do you find the derivative of #F(x) = sqrt( (x-8)/(x^2-2) )#? How do you use the chain rule to differentiate #2^(-9z^2+3z+5)#? How do you differentiate #y = x(1 - x)^2(x + 2)^3#? How do you find the derivative of #f(x) = tan(sinx)#? How do you differentiate #f(x)=x/cotsqrtx# using the chain rule? How do you find the derivative of #arctan sqrt [ (1-x)/(1+x)]#? How do you find the derivative of #f(x) = (x^3-3)/x#? How do you find the first and second derivative of #h(x)=sqrt(x^2+1)#? How do you find the derivative of # ln(x^3)#? If #f(x) =xe^x# and #g(x) = e^(3x)#, what is #f'(g(x)) #? How do you find the derivative of #f(x) = tan^2(x)#? How do you differentiate #f(x)=x/arcsinsqrt(ln(1/x^2)# using the chain rule? How do you differentiate #y=2^(3^(x^2))#? Decimal representation worksheets. How do you differentiate # y=sec (x^2/pi - xpi)# using the chain rule? How do you use the chain rule to differentiate #1/-sinx#? How do you use the chain rule to differentiate #y = e^lnx#? How do you use the chain rule to differentiate #y=(4x+5)^5#? What is the derivative of #T(w)=cot^3(3w+1)#? How do you use the chain rule to differentiate #y=sqrt(-x^4-1)(-x-2)#? How do you differentiate #f(x)=(x^2+1)^3 # using the chain rule? How do you differentiate #f(x)=sqrt(csc(2/x ) # using the chain rule? * Chain rule is used when there is only one function and it has the power. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(−2x+5)+3=−12x+30+3=−12… How do you differentiate #f(x)=ln(x^2-sqrt(2x+8)))# using the chain rule? How do you find f" given #f(x)= (6x + 5)^(1/3)#? How do you differentiate #f(x) = sin(sqrt(arcsinx)) # using the chain rule? How do you use the chain rule to differentiate #sin(e^(6x))#? How do you differentiate #f(x)=tan(e^(x-x^2)) # using the chain rule? How do you differentiate #f(x)=sqrtsin(e^(4x))# using the chain rule.? ... BODMAS Rule. How do you differentiate #g(t)=1/(t^4+1)^3#? In differential calculus, we use the Chain Rule when we have a composite function. Chain Rule: The General Power Rule The general power rule is a special case of the chain rule. If #f(x)= - x^2 + x # and #g(x) = sqrtx + x #, how do you differentiate #f(g(x)) # using the chain rule? Let f be a function of g, which in turn is a function of x, so that we have f(g(x)). If #f(x) = 4x -2# and #g(x) = e^(3x-1)#, what is #f'(g(x)) #? What is the derivative of? How do you differentiate # f(x)=e^sqrt(1/x^2)# using the chain rule.? How do you differentiate # f(x)= (6e^(-x)+2)^3 # using the chain rule? If #f(x)= (5x -1)^3-2 # and #g(x) = e^x #, what is #f'(g(x)) #? How do you differentiate #r/(r^2 + 1)^(1/2)#? What is the derivative of #(4-x^(2/3))^(3/2)#? How do you differentiate #[x^(sin3x)^2]#? How do you find the derivative of #sqrt(x - 2)#? How do you differentiate # y =cos^3(5x^2-2)# using the chain rule? How do you find the derivative of #h(x)=f(x^2)# using the chain rule? The chain rule can be tricky to apply correctly, especially since, with a complicated expression, one might need to use the chain rule multiple times. How do you use the chain rule to differentiate #y=cos((4x)^3)#? How do you differentiate #f(x)=e^((x^2+x^(1/2))^(1/2) )# using the chain rule? and any corresponding bookmarks? And this is because the derivative of e to the x if you'll recall derivative of … How do you differentiate #f(x)=sec^2(3x ) # using the chain rule? How do you differentiate #f(x)=1/(ln(1-(e^(-cos(x^2)))))^(3/2)# using the chain rule? How do you differentiate # y= sqrt( (x^2 + 4x + 1)^2+2x)# using the chain rule? In applying the Chain Rule, think of the opposite function f °g as having an inside and an outside part: General Power Rule a special case of the Chain Rule. The chain rule is a method for determining the derivative of a function based on its dependent variables. If a composite function r( x) is defined as. How do you differentiate # f(x)=(1-e^x)^2# using the chain rule.? y = tan^4(3x)#? How do you differentiate #f(x)=csc(5x^5)#? If #f(x)= - e^(5x # and #g(x) = 2x^3 #, how do you differentiate #f(g(x)) # using the chain rule? How do you use the chain rule to differentiate #y=sec2x^4#? How do you find the derivative of #y= sin{cos^2(tanx)}#? It is useful when finding the derivative of a function that is raised to the nth power. How do you differentiate #h(t)=(t^4-1)^3(t^3+1)^4#? How do you differentiate #f(x)=x/(2^sqrt(x-3))# using the chain rule? What is the derivative of #f(x)=(pi/x^5)(1/(e^(1/x)-1))#? How do you use the chain rule to differentiate #y=(6-2x)^3#? How do you differentiate #e^((ln2x)^2) # using the chain rule? Do not use substitution such as #u=3^x#. What is the derivative of #(ln(sin(2x)))^2#? How do you differentiate #f(x)=sqrt(sine^(4x)# using the chain rule.? How do you find the second derivative of #y=1/x#? Can you explain how the chain rule work in real life? So, for example, (2x +1)^3. How do you differentiate #f(x) = (4x^3 + 2x) ^ - 4#? How do you find the derivative of # pi^(x+2)# using the chain rule? Let u = 5x (therefore, y = sin u) so using the chain rule. How do you differentiate # 3/4 * (2x^3 + 3x)^(-1/4)#? How do you differentiate #f(x) = sin(xcos(x))# using the chain rule? Is there a chain rule for partial derivatives? How do you differentiate #f(x)=sin^2(lnx)xcos^2(x^2)# using the chain rule? How do you differentiate # y =sqrt((3x-9)^3 # using the chain rule? What is the second derivative of #f(cosx)# when #x=pi/2# where #f(x)=sinx#? How do you find the second derivative of #y=lnx#? Using the chain rule: Because the argument of the sine function is something other than a plain old x, this is a chain rule problem. How do you differentiate #arcsin(csc(4/x)) )# using the chain rule? How do you find the derivative of #((sinx)^2)/(1-cosx)#? How do you find the derivative of #y= (4x-x^2)^10# ? How do you differentiate #f(x)=2 ln( x^2 - 3x +4) # using the chain rule? How do you differentiate #f(x)=(sin2x^2)/4# using the chain rule? If #f(x)= 1/x # and #g(x) = 1/x #, how do you differentiate #f'(g(x)) # using the chain rule? How do you differentiate #y=sqrt(t/(t^2+4))#? How do you find the derivative of #f(x)=-xe^x + 2#? How do you differentiate #f(x)=cos(-3x^2+2)^2#? How do you find dy/dx given #y=ln(2+x^2)#? How do you find the derivative of #xsqrt(2x - 3)#? If #f(x) =sec^2(x/2) # and #g(x) = sqrt(5x-1 #, what is #f'(g(x)) #? How do you find the derivative of #(x^2 + 1/x)^5#? How do you find the derivative of #cos(x^2)#? The chain rule says that. What is the derivative of #(4x)^3 * (2x)^6#? Chain rule is also often used with quotient rule. Basic examples that show how to use the chain rule to calculate the derivative of the composition of functions. How do you differentiate #f(x)=sqrt(tane^(4x)# using the chain rule.? How do you differentiate #f(t)=ln(e^(sin^2t))# using the chain rule? How do you find the fourth derivative of #e^(-x)#? How do you find #(dy)/(dx)# given #x^3+y^3=3xy^2#? How do you find the first and second derivative of #y=e^(e^x)#? from your Reading List will also remove any How do you differentiate #f(x) = sin(1/sqrt(arcsinx)) # using the chain rule? How do you find the derivative of #q(x) = (8x) ^(2/3)# using the chain rule? How do you use the chain rule to differentiate #y=1/(2x^3+7)#? What is the derivative of # (ln (x-4)) ^ 3#? If #f(x)= 3x^3-2 # and #g(x) = e^x #, what is #f'(g(x)) #? How do you differentiate #f(x)=-3tan4x^2# using the chain rule? Let's see what that looks like mathematically: Let's say we have the composite function #sin(5x)#. How do you use the chain rule to differentiate #f(x)=(3x-9)^2(4x^3+2x^-9)^-9#? How do you differentiate #y=[x(x+sin^2x)^3]^4#? How do you differentiate #f(x)=sqrtcos(1/(2x)^3)# using the chain rule? How do you differentiate #f(x)=(x^2+x^(1/2))^(1/2) # using the chain rule? How do you differentiate #y = ln [x^4 sin^2 (x)]#? How do you differentiate # f(x)=xsqrt(3x-e^x)# using the chain rule.? How do you differentiate #f(x)=(ln(sinx)^2-3xln(sinx)+x^2ln(cos^2x^2)# using the chain rule? How do you differentiate # f(x)=e^((6x-2)^2 # using the chain rule.? How do you differentiate # f(x)=1/sqrt((7-2x^3)# using the chain rule.? How do you differentiate # f(x)=ln(6x+8)# using the chain rule.? A common interpretation is to multiply the rates: x wiggles f. This creates a rate of change of df/dx, which wiggles g … if #v=1148sqrt(p)# where #p# is a function of #t# then find #(dv)/dt# when #p=44.0# and #dp/dt=0.307#? How do you differentiate #f(x)=x/ln(sqrt(1/x))# using the chain rule? How do you find the derivative of #g(t) = 1/t^(1/2)#? How do you find the derivative of #tan^2x#? The following problems require the use of the chain rule. How do you use the chain rule to differentiate #cos(10csc10x)#? How do you differentiate #f(x) = (−7 x^2 − 5)^8 (2 x^2 − 9)^9# using the chain rule? How do you differentiate #f(x) = sqrt[ (3 x + 1) / (5 x^2 + 1)# using the chain rule? How do you differentiate #f(x)=(ln(sinx)^2/(x^2ln(cos^2x^2)# using the chain rule? How do you differentiate # f(x)=(ln(6x+8))^2# using the chain rule.? How do you differentiate # y=ln((6x-5)^6) # using the chain rule? How do you find the derivative of #y= x*sin(1/x)# ? How do you differentiate # y= -2( x^2 + 1 )^2# using the chain rule? How do you differentiate #f(x)=e^(x^2+x+8) # using the chain rule? How do you differentiate # f(t)=sin^2(e^(sin^2t)# using the chain rule.? How do you differentiate #f(x) = xcos((pix)/2)# using the chain rule? How do you differentiate #f(x)=cot(1/e^x) # using the chain rule? How do you find the derivative of #f(x) = 4e^(3x+2)#? If #f(x) = -x -2# and #g(x) = e^(x^2-x)#, what is #f'(g(x)) #? How do you find the derivative of #y=sqrt( x+sqrt( x+sqrt( x)))#? How do you differentiate #f(x)=-5 xe^(x/cos x)# using the chain rule? The chain rule is a rule for differentiating compositions of functions. People ask this question all the time. How do you find the derivative of #ln(1+e^x)#? How do you find the derivative of #cos(pi/2)#? {eq}\displaystyle y = (2x^2 - 3)^3 {/eq} Chain and Power Rule: The given function is in the form of the composition of function. What is the derivative of #sqrt(4+x^2) #? How do you use the chain rule to differentiate #y=7/(2x+7)^2#? How do you differentiate #f(x)=cos(x^2-4x) # using the chain rule? How do you differentiate # f(x)=ln(1/sqrt(e^x-x))# using the chain rule.? How do you differentiate # f(t)=-e^(sin(pi/x))sinpix # using the chain rule.? How do you differentiate #f(t)= sqrt(( 1+ ln(t) ) / ( 1 - ln(t) ) #? How do you find the derivative of #f(x) = x + x^(1/2)#? How do you find the derivative of #y = sqrt(2x - x^2)#? How do you use the chain rule to differentiate #y=2(x+3)^(1/2)#? How do you find the derivative of #sqrt(x^2-1)#? How do you determine #(dy)/(dx)# given #y=cos(1-x)#? How do you use the chain rule to differentiate #f(x)=(x^7-3x^2+15x^(-3/2))^6#? How do you differentiate #f(x)=sqrt(e^(cot(1/x)# using the chain rule.? How do you find the derivative of # f(t)=sin^2[e^(sin^2)t]# using the chain rule? How do you use the chain rule to differentiate #y=(x^2+1)^(1/2)#? All functions are functions of real numbers that return real values. What is the derivative of #w =sqrt(x^2+y^2+z^2)#? How do you differentiate #sin(x^2)(cos(x^2))#? If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of each function given below. How do you find the derivative of #(x)/sqrt(x^2-4)#? How do you differentiate #y=(4/ln2)(x^(lnx))(lnx+2ln(cosx))#? How do you differentiate #y = 2 / [3sqrt(x^2 - 5x)] #? How do you differentiate #f(x) = ln(sqrt(arcsin(e^(2-x)) ) # using the chain rule? How do you differentiate #f(x)=(cot(x))^2 # using the chain rule? How do you differentiate #3sin^3(2x^2) # using the chain rule? Using the point-slope form of a line, an equation of this tangent line is or . Proof of the chain rule. If #f(x) =sin^3x # and #g(x) = sqrt(3x-1 #, what is #f'(g(x)) #? How do you use the chain rule to differentiate #y=(3x^3+1)(-4x^2-3)^4#? What is the derivative of #ln[(x(x^2+1)^2)/(2x^3-1)^(1/2)] #? How do you differentiate #f(x)=tan(1-3x) # using the chain rule? How do you differentiate #y=(2x-5)^4(8x^2-5)^-3#? How do you differentiate #f(x)=(1/(x-3)^2)^2# using the chain rule? How do you differentiate #f(x)=4x ln(3sin^2x^2 + 2)# using the chain rule? How do you find the derivative of #sqrt((x^2-1) / (x^2+1))#? How do you differentiate #f(x)=sqrtcsc(2x -4) # using the chain rule? How do you differentiate #f(x)=csc(e^(x^3-x)) # using the chain rule? Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. How do you use the chain rule to differentiate #root3(4x+9)#? How do you find the derivative of the function #y = sin(tan(5x))#? Example of Chain Rule. If #f(x) =sec^3(x/2) # and #g(x) = sqrt(2x-1 #, what is #f'(g(x)) #? If #f(x) =xe^x# and #g(x) = sin3x #, what is #f'(g(x)) #? How do you differentiate #f(x)=ln(cos(e^(x) )) # using the chain rule? How do you differentiate #f(x)=sin(1/sqrt(3x^2-4) ) # using the chain rule? How do you differentiate #f(x)=(2x-3)^3# using the chain rule? How do you differentiate #f(x)=sin e^(4x)# using the chain rule.? If (below) is f'(x) then what is f(x)? This line passes through the point . The chain rule lets us "zoom into" a function and see how an initial change (x) can effect the final result down the line (g). If #f(x)= sin3x # and #g(x) = 2x^2 #, how do you differentiate #f(g(x)) # using the chain rule? #y=((1+x)/(1-x))^3=((1+x)(1-x)^-1)^3=(1+x)^3(1-x)^-3#. How do you differentiate #arcsin(csc(1-1/x^3)) )# using the chain rule? How do you find the derivative of # y=x^3(7x-7)^5# using the chain rule? By the way, here’s one way to quickly recognize a composite function.
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