## Continuity and Differentiability

# Logarithmic Differentiation

- Logarithmic differentiation is a powerful technique to differentiate functions of the form
*f(x)*= [*u(x)*]^{v(x)}. Here both*f(x)*and*u(x)*need to be positive for this technique to make sense.

### View the Topic in this video From 32:43 To 49:02

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- \frac{d}{dx}\left(f(x)^{g(x)}\right)=f(x)^{g(x)}\left[g(x)\frac{f'(x)}{f(x)}+g'(x) \log \ f(x)\right]
- \frac{d}{dx}\left(f(x)^{f(x)}\right)=f(x)^{f(x)}\left[(1+ \log \ f(x))f'(x)\right]
- If y=(f(x))^y \Rightarrow \frac{dy}{dx}=\frac{y^{2}f'(x)}{f(x)[1-y \ \log(f(x))]}
- \frac{d}{dx}\left(\cos^{n}x \ \cos \ nx\right)=n\cos^{n-1}x \ \cos (n+1)x
- \frac{d}{dx}\left(\sin^{n}x \ \sin \ nx\right)=n\sin^{n-1}x \ \sin (n+1)x
- \frac{d}{dx}\left(\cos^{n}x \ \sin \ nx\right)=-n\cos^{n-1}x \ \sin (n+1)x
- \frac{d}{dx}\left(\sin^{n}x \ \cos \ nx\right)=-n\sin^{n-1}x \ \cos (n+1)x
- \frac{d}{dx}\left(Variable^{Variable}\right)=\frac{d}{dx}(Variable)^{Constant}+\frac{d}{dx}(Constant)^{Variable}