Question

Derivative of the function $f(x) = log_5(log_7x)$, $x > 7$ is

• A. $\frac{1}{x\left(log\,5\right)\left(log\,7\right)\left(log_{7}\,x\right)}$
• B. $\frac{1}{x\left(log\,5\right)\left(log\,7\right)}$
• C. $\frac{1}{x\left(log\,x\right)}$
• D. None of these

Answer 1

Answer: (A)

$\frac{1}{x\left(log\,5\right)\left(log\,7\right)\left(log_{7}\,x\right)}$

Answer 2

$f \left(x\right)=log_{5}\left(log_{7}x\right)=log_{5}\left(\frac{log_{e}\,x}{log_{e}\,7}\right)$ $=log_{5}\left(log_{e}\,x\right)-log_{5}\left(log_{e}7\right)$ $=\frac{log_{e}\left(log_{e}\,x\right)}{log_{e}\,5}-log_{5}\left(log_{e}\right)\,7$ Differentiating $w$.$r$.$t$. $x$, we get $f '\left(x\right)=\frac{1}{log_{e}\,x}\cdot\frac{1}{x}\cdot\frac{1}{log_{e}\,5}$ $=\frac{1}{\frac{x\,log_{e}\,x}{log_{e}\,7}log_{e}\,7\cdot log_{e}\,5}$ $=\frac{1}{x\,log_{7}\,x\cdot log\,7\cdot log\,5}$