Find the derivative: frac{d}{dx}[(log_e x)(lo | vedclass

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

Question

(D) We need to find the derivative of the product of two logarithmic functions: $\frac{d}{dx}[(\log_e x)(\log_a x)]$. First,use the change of base for

Vedclass · Accepted Answer

$\frac{2\log_a x}{x}$

Vedclass · Answer

(D) We need to find the derivative of the product of two logarithmic functions: $\frac{d}{dx}[(\log_e x)(\log_a x)]$. First,use the change of base formula for logarithms: $\log_a x = \frac{\log_e x}{\log_e a}$. Substituting this into the expression: $\frac{d}{dx}[(\log_e x) \cdot \frac{\log_e x}{\log_e a}] = \frac{1}{\log_e a} \cdot \frac{d}{dx}[(\log_e x)^2]$. Using the chain rule,the derivative of $(\log_e x)^2$ is $2(\log_e x) \cdot \frac{d}{dx}(\log_e x) = 2(\log_e x) \cdot \frac{1}{x}$. Thus,the expression becomes: $\frac{1}{\log_e a} \cdot \frac{2\log_e x}{x} = \frac{2}{\log_e a} \cdot \frac{\log_e x}{x}$. Since $\frac{\log_e x}{\log_e a} = \log_a x$,the final result is $\frac{2\log_a x}{x}$.

Find the derivative: $\frac{d}{dx}[(\log_e x)(\log_a x)]$

Similar Questions

$\frac{d}{d x}(\log _{|x|} e) =$ . . . . . .

If $y = \log_{\sin x} (\tan x)$,then $\left( \frac{dy}{dx} \right)_{\pi/4}$ is equal to

If $f(x) = \log_{(x^2-2x+1)}(x^2-3x+2)$,$x \in R - \{1, 2\}$ and $x \neq 0$,then $f'(3) =$

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

$\frac{d}{dx} \left( \log \sqrt{\frac{1+\sin x}{1-\sin x}} \right) = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module