Question:
Derivative of the function $f(x) = log_5(log_7x)$, $x > 7$ is
Derivative of the function $f(x) = log_5(log_7x)$, $x > 7$ is
Updated On: Jul 6, 2022
- $\frac{1}{x\left(log\,5\right)\left(log\,7\right)\left(log_{7}\,x\right)}$
- $\frac{1}{x\left(log\,5\right)\left(log\,7\right)}$
- $\frac{1}{x\left(log\,x\right)}$
- None of these
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The Correct Option is A
Solution and Explanation
$f \left(x\right)=log_{5}\left(log_{7}x\right)=log_{5}\left(\frac{log_{e}\,x}{log_{e}\,7}\right)$
$=log_{5}\left(log_{e}\,x\right)-log_{5}\left(log_{e}7\right)$
$=\frac{log_{e}\left(log_{e}\,x\right)}{log_{e}\,5}-log_{5}\left(log_{e}\right)\,7$
Differentiating $w$.$r$.$t$. $x$, we get
$f '\left(x\right)=\frac{1}{log_{e}\,x}\cdot\frac{1}{x}\cdot\frac{1}{log_{e}\,5}$
$=\frac{1}{\frac{x\,log_{e}\,x}{log_{e}\,7}log_{e}\,7\cdot log_{e}\,5}$
$=\frac{1}{x\,log_{7}\,x\cdot log\,7\cdot log\,5}$
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Notes on Continuity and differentiability
Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.