Logarithmic differentiation

Tags

Calculus

Cegep/1

Word count

369 words

Reading time

3 minutes

Used to simplify differentiation or differentiate otherwise impossible equations

0. Preferably convert explicit differentiation to implicit by replacing $f(x)$ by $y$
1. Apply $\ln$
2. Simplify
3. Differentiate
4. Solve for the derivative

Examples

Find $y^{\prime }$ when $y=x^x$.

$$\begin{array}{rl}y& =x^x\\ ⟹\ln y& =\ln x\\ & =x\ln x\\ ⟹\frac{\mathrm{d}}{\mathrm{d}x}\ln y& =\frac{\mathrm{d}}{\mathrm{d}x}(x\ln x)\\ \frac{y^{\prime }}{y}& =\ln x+x\cdot \frac{1}{x}\\ & =\ln x+1\\ y^{\prime }& =y(\ln x+1)\\ & =x^x(\ln x+1)\end{array}$$

Find $y^{\prime }$ when $y=x^{x^x}$.

$$\begin{array}{rl}y& =x^{x^x}\\ ⟹\ln y& =\ln x^{x^x}\\ \ln y& =x^x\ln x\\ ⟹\ln \ln y& =\ln (x^x\ln x)\\ & =\ln x^x+\ln \ln x\\ & =x\ln x+\ln \ln x\\ \frac{\mathrm{d}}{\mathrm{d}x}\ln \ln y& =\frac{\mathrm{d}}{\mathrm{d}x}(x\ln x+\ln \ln x)\\ \frac{(\ln y{)}^{\prime }}{\ln y}& =\ln x+x\cdot \frac{1}{x}+\frac{(\ln x{)}^{\prime }}{\ln x}\\ \frac{y^{\prime }}{y\ln y}& =\ln x+1+\frac{x^{\prime }}{x\ln x}\\ y^{\prime }& =y\ln y(\ln x+1+\frac{1}{x\ln x})\\ & =x^{x^x}\ln x^{x^x}(\ln x+1+\frac{1}{x\ln x})\end{array}$$

Find the derivative of $f(x)=\sqrt[3]{\frac{\sqrt{x}{e}^{x^2}(x^7+2{)}^{-3}}{{\cos }^{2}x}}$.

$$\begin{array}{rl}y& =\sqrt[3]{\frac{\sqrt{x}{e}^{x^2}(x^7+2{)}^{-3}}{{\cos }^{2}x}}\\ ⟹\ln y& =\ln \sqrt[3]{\frac{\sqrt{x}{e}^{x^2}(x^7+2{)}^{-3}}{{\cos }^{2}x}}\\ & =\frac{1}{3}(\ln (\sqrt{x}{e}^{x^2}(x^7+2{)}^{-3})-\ln {\cos }^{2}x)\\ & =\frac{1}{3}(\ln \sqrt{x}+\ln e^{x^2}+\ln (x^7+2{)}^{-3}-\ln {\cos }^{2}x)\\ & =\frac{1}{3}(\ln \sqrt{x}+x^2-3\ln (x^7+2)-\ln {\cos }^{2}x)\\ ⟹\frac{\mathrm{d}}{\mathrm{d}x}\ln y& =\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}(\ln \sqrt{x}+x^2-3\ln (x^7+2)-\ln {\cos }^{2}x))\\ \frac{y^{\prime }}{y}& =\frac{1}{3}(\frac{{\sqrt{x}}^{\prime }}{\sqrt{x}}+2x-\frac{3(x^7+2{)}^{\prime }}{x^7+2}-\frac{({\cos }^{2}x{)}^{\prime }}{{\cos }^{2}x})\\ & =\frac{1}{3}(\frac{\frac{1}{2}{x}^{-1/2}}{\sqrt{x}}+2x-\frac{21x^6}{x^7+2}+\frac{2\cos x\sin x}{{\cos }^{2}x})\\ y^{\prime }& =\sqrt[3]{\frac{\sqrt{x}{e}^{x^2}(x^7+2{)}^{-3}}{{\cos }^{2}x}}\frac{1}{3}(\frac{\frac{1}{2}{x}^{-1/2}}{\sqrt{x}}+2x-\frac{21x^6}{x^7+2}+\frac{2\cos x\sin x}{{\cos }^{2}x})\end{array}$$

Contributors

Ajitani Hifumi

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