Question:
If $g$ is the inverse function of $f$ and $f '(x) = sin\, x$, then $g '(x)$ is
If $g$ is the inverse function of $f$ and $f '(x) = sin\, x$, then $g '(x)$ is
Updated On: Jul 6, 2022
- $cosec\left\{g\left(x\right)\right\}$
- $sin\left\{g\left(x\right)\right\}$
- $-\frac{1}{sin\left\{g\left(x\right)\right\}}$
- $cos\left\{g\left(x\right)\right\}$
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The Correct Option is A
Solution and Explanation
Given $f^{ -1}(x) = g(x)$
$\Rightarrow x=f \left[g\left(x\right)\right]$
Diff. both side $w.r.t \left(x\right)$
$\Rightarrow 1=f '\left[g\left(x\right)\right].g '\left(x\right) \Rightarrow g '\left(x\right)=\frac{1}{f '\left(g\left(x\right)\right)}$
Given, $f '\left(x\right) = sin\, x$
$\therefore f '\left(g\left(x\right)\right)=sin\left[g\left(x\right)\right]$
$\Rightarrow \frac{1}{f '\left(g\left(x\right)\right)}=co\,sec\left[g\left(x\right)\right]$
Hence, $g '\left(x\right) = cosec\left[g\left(x\right)\right]$
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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.