The derivative of the function f(x) = log_5(l | vedclass

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Question

(A) Given the function $f(x) = \log_5(\log_7 x)$.Using the change of base formula $\log_a b = \frac{\ln b}{\ln a}$,we can write:$f(x) = \frac{\ln(\log

Vedclass · Accepted Answer

$\frac{1}{x(\ln 5)(\ln 7)(\log_7 x)}$

Vedclass · Answer

(A) Given the function $f(x) = \log_5(\log_7 x)$.Using the change of base formula $\log_a b = \frac{\ln b}{\ln a}$,we can write:$f(x) = \frac{\ln(\log_7 x)}{\ln 5}$$f(x) = \frac{\ln(\frac{\ln x}{\ln 7})}{\ln 5}$$f(x) = \frac{\ln(\ln x) - \ln(\ln 7)}{\ln 5}$Now,differentiating with respect to $x$:$f'(x) = \frac{1}{\ln 5} \cdot \frac{d}{dx}(\ln(\ln x) - \ln(\ln 7))$$f'(x) = \frac{1}{\ln 5} \cdot \frac{1}{\ln x} \cdot \frac{d}{dx}(\ln x)$$f'(x) = \frac{1}{\ln 5} \cdot \frac{1}{\ln x} \cdot \frac{1}{x}$$f'(x) = \frac{1}{x(\ln 5)(\ln x)}$Since $\ln x = \ln 7 \cdot \log_7 x$,we have:$f'(x) = \frac{1}{x(\ln 5)(\ln 7)(\log_7 x)}$Thus,the correct option is $A$.

The derivative of the function $f(x) = \log_5(\log_7 x)$ for $x > 7$ is:

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