Kelly Hardy
01/11/2023 · Primary School
\( f(x)=\frac{\ln \left(2 x^{4}\right)}{\tan x} \), find \( f^{\prime}(\mathrm{x}) \) \( \frac{4 \tan x-x \ln \left(2 x^{4}\right) \sec ^{2} x}{x \tan ^{2} x} \) \( \frac{2 \ln 2 \tan x-x \ln \left(2 x^{4}\right) \sec ^{2} x}{x \tan ^{2} x} \) \( \frac{4 \tan x+x \ln \left(2 x^{4}\right) \sec ^{2} x}{x \tan ^{2} x} \) \( \frac{2 \ln 2 \tan x+x \ln \left(2 x^{4}\right) \sec ^{2} x}{x \tan ^{2} x} \)
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Tutor-Verified Answer
Step-by-step Solution
Find the first order derivative with respect to \( x \) for \( \frac{\ln \left(2 x^{4}\right)}{\tan x} \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\frac{\ln{\left(2x^{4}\right)}}{\tan\left(x\right)}\right)\)
- step1: Use differentiation rules:
\(\frac{\frac{d}{dx}\left(\ln{\left(2x^{4}\right)}\right)\times \tan\left(x\right)-\ln{\left(2x^{4}\right)}\times \frac{d}{dx}\left(\tan\left(x\right)\right)}{\tan^{2}\left(x\right)}\)
- step2: Calculate:
\(\frac{\frac{4}{x}\times \tan\left(x\right)-\ln{\left(2x^{4}\right)}\times \sec^{2}\left(x\right)}{\tan^{2}\left(x\right)}\)
- step3: Calculate:
\(\frac{\frac{4\tan\left(x\right)}{x}-\ln{\left(2x^{4}\right)}\times \sec^{2}\left(x\right)}{\tan^{2}\left(x\right)}\)
- step4: Calculate:
\(\frac{\frac{4\tan\left(x\right)-x\ln{\left(2x^{4}\right)}\times \sec^{2}\left(x\right)}{x}}{\tan^{2}\left(x\right)}\)
- step5: Multiply by the reciprocal:
\(\frac{4\tan\left(x\right)-x\ln{\left(2x^{4}\right)}\times \sec^{2}\left(x\right)}{x}\times \frac{1}{\tan^{2}\left(x\right)}\)
- step6: Multiply the terms:
\(\frac{4\tan\left(x\right)-x\ln{\left(2x^{4}\right)}\times \sec^{2}\left(x\right)}{x\tan^{2}\left(x\right)}\)
- step7: Multiply the terms:
\(\frac{4\tan\left(x\right)-x\ln{\left(2x^{4}\right)}\times \sec^{2}\left(x\right)}{\tan^{2}\left(x\right)\times x}\)
The first derivative of the function \( f(x)=\frac{\ln \left(2 x^{4}\right)}{\tan x} \) with respect to \( x \) is:
\[ f'(x) = \frac{4\tan(x) - x\ln(2x^{4})\sec^{2}(x)}{x\tan^{2}(x)} \]
Therefore, the correct option is \( \frac{4\tan x-x \ln \left(2 x^{4}\right) \sec ^{2} x}{x \tan ^{2} x} \).
Quick Answer
The correct option is \( \frac{4\tan x-x \ln \left(2 x^{4}\right) \sec ^{2} x}{x \tan ^{2} x} \).
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