Best solutions to - "Find the derivatives of function (log⁡x)tan⁡x" - Plainmath

Phoebe

Answered question

2021-10-06

Find the derivatives of function

{(\log x)}^{\tan x}

Answer & Explanation

crocolylec

Skilled2021-10-07Added 100 answers

Step 1
Given
Let

y = {(\log x)}^{\tan x}

Step 2 Finding the derivatives
Taking logarithms on both sides

\log y = \log [{(\log x)}^{\tan x}]

\log y = \tan x \cdot \log (\log x)

Differentiating w.r.t x

\frac{1}{y} \frac{d y}{d x} = \tan x \frac{d}{d x} \log (\log x) + \log (\log x) \frac{d}{d x} \tan x

\frac{1}{y} \frac{d y}{d x} = \tan x \frac{1}{\log x} \frac{1}{x} + \log (\log x) \sec^{2} x

\frac{d y}{d x} = y [\frac{\tan x}{x \log x} + \log (\log x) \sec^{2} x]

∴ \frac{d y}{d x} = {(\log x)}^{\tan x} [\frac{\tan x}{x \log x} + \log (\log x) \cdot \sec^{2} x]

Jeffrey Jordon

Expert2022-03-13Added 2605 answers

Answer is given below (on video)

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