# 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=loge] on both sides, we get that

ln y = sin-1x as we know that ln ek =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 esin-1x 1/√(1-x2).

Derivative of arc(cotx): The derivative of arc(cotx) is -1/(1+x2).

Differentiate esinx: The derivative of esinx is esinx cosx.

Differentiate ecosx: The derivative of ecosx is -ecosx sinx.

Differentiate etanx: The derivative of etanx is etanx sec2x.

## FAQs

### Q1: What is the derivative of e to the power sin inverse x?

Answer: The derivative of e to the power sin-1x is equal to esin-1x 1/√(1-x2).

### Q2: If y=esin-1x, then find dy/dx?

Answer: If y=esin-1x, then dy/dx = esin-1x 1/√(1-x2).