Check the solution to "Evaluate the derivatives of the following functions. f(x)=sin−1(esin⁡x)" - Plainmath

aortiH

Answered question

2021-09-23

Evaluate the derivatives of the following functions.

f (x) = \sin^{- 1} (e^{\sin x})

Answer & Explanation

saiyansruleA

Skilled2021-09-24Added 110 answers

Step 1

\frac{d}{d x} [\arcsin (e^{\sin (x)})]

= \frac{1}{\sqrt{1 - {(e^{\sin x})}^{2}}} \cdot \frac{d}{d x} [e^{\sin x}]

= \frac{e^{\sin x} \cdot \frac{d}{d x} [\sin x]}{\sqrt{1 - e^{2 \sin (x)}}}

= \frac{e^{\sin x} \cos (x)}{\sqrt{1 - e^{2 \sin (x)}}}

Step 2

f (x) = \sin^{- 1} (e^{\sin (x)})

then,

f^{'} (x) = \frac{e^{\sin x} \cos (x)}{\sqrt{1 - e^{2 \sin (x)}}}

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