If f(x)=log _eleft(e^{2 x}left(frac{3 x+5}{5- | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given, $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{\frac{2}{3}}\right)$ Using the property $\log(ab) = \log a + \log b$, we get: $

Vedclass · Accepted Answer

$\frac{13}{4}$

Vedclass · Answer

(D) Given, $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{\frac{2}{3}}\right)$ Using the property $\log(ab) = \log a + \log b$, we get: $f(x)=\log _e e^{2 x}+\log _e\left(\frac{3 x+5}{5-3 x}\right)^{\frac{2}{3}}$ $f(x)=2 x+\frac{2}{3} \log _e\left(\frac{3 x+5}{5-3 x}\right)$ Using $\log(\frac{a}{b}) = \log a - \log b$: $f(x)=2 x+\frac{2}{3}\left(\log _e(3 x+5)-\log _e(5-3 x)\right)$ Differentiating with respect to $x$: $\frac{d f}{d x}=2+\frac{2}{3}\left(\frac{3}{3 x+5}-\frac{-3}{5-3 x}\right)$ $\frac{d f}{d x}=2+2\left(\frac{1}{3 x+5}+\frac{1}{5-3 x}\right)$ At $x=1$: $\left(\frac{d f}{d x}\right)_{x=1}=2+2\left(\frac{1}{3(1)+5}+\frac{1}{5-3(1)}\right)$ $=2+2\left(\frac{1}{8}+\frac{1}{2}\right)=2+2\left(\frac{1+4}{8}\right)=2+2\left(\frac{5}{8}\right)=2+\frac{5}{4}=\frac{13}{4}$

If $f(x)=\log _e\left(e^{2 x}\left(\frac{3 x+5}{5-3 x}\right)^{\frac{2}{3}}\right)$, $x \neq \frac{-5}{3}, \frac{5}{3}$, then the value of $\frac{d f}{d x}$ at $x=1$ is

Similar Questions

Find the derivative: $\frac{d}{dx}(\log_7(\log_7 x))$

If $y = \log_{10}(x^2)$,then $\frac{dy}{dx}$ is equal to

$\frac{d}{dx} \log \tan \left( \frac{\pi}{4} + \frac{x}{2} \right) = $

If $y = e^{(1 + \log_e x)}$,then the value of $\frac{dy}{dx} = $

If $y = 3^{x^2}$,then $\frac{dy}{dx}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module