Differentiate e to power sin inverse x | Differentiation of e^sin^-1x

The differentiation of e to the sin^-1x is equal to 1/√(1-x²) e^{sin^-1x}. In this post, we will learn how to differentiate e to power sin inverse x with respect to x.

Let us find the derivative of e^{sin^-1x} with respect to x.

Find the derivative of e^{sin^-1x}

Question: How to differentiate e^{sin^-1x}?

Solution:

Let y= $e^{\sin^{-1}x}$

Taking natural logarithms [ln=log_e] on both sides, we get that

ln y = sin^-1x as we know that ln e^k =k.

Differentiate both sides with respect to x. This will give us

$\dfrac{1}{y} \dfrac{dy}{dx}$ = $\dfrac{1}{\sqrt{1-x^2}}$, because the derivative of sin^-1x is 1/√(1-x²).

⇒ $\dfrac{dy}{dx}$ = $\dfrac{y}{\sqrt{1-x^2}}$

Putting the value of y, that is, y=e^{sin^-1x}, we get that

$\dfrac{d}{dx} \Big(e^{\sin^{-1}x} \Big)$ = $e^{\sin^{-1}x} \dfrac{1}{\sqrt{1-x^2}}$.

Thus, the differentiation of e to the power sin inverse x with respect to x is equal to e^sin-1x 1/√(1-x²).

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Differentiate e^cosx: The derivative of e^cosx is -e^cosx sinx.

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