# Differentiation of e^cos^-1x

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

We will now find the derivative of e^{cos-1x} with respect to x.

## Find the derivative of e^{cos-1x}

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

Solution:

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

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

ln y = cos-1x as we know that ln(ek) =k.

Differentiating both sides with respect to x, we obtain that

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

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

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

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

Thus, the differentiation of e to the power cos inverse x with respect to x is equal to -ecos-1x 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.

Differentiate e^{sin-1x}: The derivative of e to the power sin inverse x is esin-1x 1/√(1-x2).

## FAQs

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

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

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

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