Variable power
\[\frac{d}{dx}\,u^v=u^v\left(v'\log u+\frac{v\,u'}{u}\right)\]
Continuity and Differentiability
Logarithmic Differentiation
Understand
For functions of the form \(y=[u(x)]^{v(x)}\) (variable base and variable exponent), or long products and quotients, take \(\log\) of both sides first:
\[\log y=v\log u\;\Rightarrow\;\frac1y\frac{dy}{dx}=v'\log u+v\,\frac{u'}{u}\]
Example: \(y=x^x\Rightarrow\log y=x\log x\Rightarrow\frac{y'}{y}=\log x+1\Rightarrow y'=x^x(1+\log x)\).
For a sum like \(y=x^x+(\sin x)^x\), differentiate each term separately by this method – do not take the logarithm of a sum.
Key Concepts
- Use when both base and exponent vary.
- Never apply log to a sum; split into u + v first.
- Multiply back by y at the end.
Formula Bank
Common Mistakes
Treating xˣ as a power or an exponential
\(\frac{d}{dx}x^x\neq x\cdot x^{x-1}\) and \(\neq x^x\log x\). Correct: \(x^x(1+\log x)\).
Solved Examples
Differentiating (sin x)ˣ
\(y=(\sin x)^x\).
\(\log y=x\log\sin x\Rightarrow\frac{y'}{y}=\log\sin x+x\cot x\Rightarrow y'=(\sin x)^x(\log\sin x+x\cot x)\).
Practice & Topic Test
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Take logs, differentiate implicitly, multiply by y. Treat each term of a sum separately.
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