Question:
If $y=\left(tanx\right)^{sinx}$, then $\frac{dy}{dx}$ is equal to
If $y=\left(tanx\right)^{sinx}$, then $\frac{dy}{dx}$ is equal to
Updated On: Jul 6, 2022
- $sec\, x + cos\, x$
- $sec\, x +log\, tan \,x$
- $(tan\,x)sin^x$
- None of these
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The Correct Option is D
Solution and Explanation
We have, $y = (tanx)^{sinx}$
Taking log on both sides, we get
$logy = sinx \,log(tanx)$
Differentiating $w$.$r$.$t$. $x$, we get
$\frac{1}{y} \frac{dy}{dx}=\frac{sin\,x}{tan\,x}\cdot sec^{2}x+cosx\,log\left(tanx\right)$
$=\left(tanx\right)^{sinx}\left[sec\,x+cosx\left(log\,tan\,x\right)\right]$
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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.