Wize AP Calculus (BC) Textbook > Differentiation 2: Composite, Implicit, & Inverse Functions

Bonus: Logarithmic Differentiation (Important technique in calculus but not usually covered in AP Calculus AB or BC)

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Logarithmic Differentiation
Consider a function of the form y=g(x)f(x).y=g(x)^{f(x)}.y=g(x)f(x). To compute the derivative of this function, it is useful (and sometimes necessary) to first take the logarithm on both sides of the equation and then compute the derivative. This process is called logarithmic differentiation.
Procedure for Logarithmic Differentiation
Take the natural logarithm on both sides of the equation: ln⁡y=ln⁡ (g(x)f(x))\ln y=\ln\ (g(x)^{f(x)})lny=ln (g(x)f(x))
Simplify using log rules
Differentiate Implicitly
Solve  
Simplify
Substitute given information (if asked to)
Wize Tip
You can use Logarithmic Differentiation on derivatives of complex products or quotients by using properties of logarithms. 

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Example: Logarithmic Differentiation
 Find the derivative of f(x)=xxf(x)=x^xf(x)=xx.

1) ln⁡f(x)=ln⁡xx\text{1)}  \ \ln f(x) = \ln x^x1) lnf(x)=lnxx

2) ln⁡f(x)=xln⁡x\text{2)} \ \ln f(x) = x \ln x2) lnf(x)=xlnx
3) f′(x)f(x)=1×ln⁡x+x×1x\text{3)} \ \displaystyle \frac{f'(x)}{f(x)}=1\times\ln x + x \times\frac{1}{x}3) f(x)f′(x)​=1×lnx+x×x1​

4) f′(x)=f(x)[ln⁡x+1]\text{4)}  \ f'(x)=f(x)[\ln x +1]4) f′(x)=f(x)[lnx+1]

5) f′(x)=xx[ln⁡x+1]\text{5)} \ f'(x)=x^x[\ln x +1]5) f′(x)=xx[lnx+1]

Find the derivative of f(x)=sin⁡xcos⁡xf(x)=\sin{x}^{\cos{x}}f(x)=sinxcosx.

−sin⁡xln⁡sin⁡x+cos⁡xcot⁡x-\sin{x}\ln{\sin{x}}+\cos{x}\cot{x}−sinxlnsinx+cosxcotx

sin⁡xcos⁡x(sin⁡xln⁡sin⁡x+cos⁡x)\sin{x}^{\cos{x}}(\sin{x}\ln{\sin{x}}+\cos{x})sinxcosx(sinxlnsinx+cosx)

sin⁡xcos⁡x(sin⁡xln⁡sin⁡x+cos⁡xcot⁡x)\sin{x}^{\cos{x}}(\sin{x}\ln{\sin{x}}+\cos{x}\cot{x})sinxcosx(sinxlnsinx+cosxcotx)

sin⁡xcos⁡x(−sin⁡xln⁡sin⁡x+cos⁡2xsin⁡x)\sin{x}^{\cos{x}}(-\sin{x}\ln{\sin{x}}+\dfrac{\cos^2{x}}{\sin{x}})sinxcosx(−sinxlnsinx+sinxcos2x​)

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