The differentiation of e^cos^-1x is equal to -1/√(1-x^{2}) e^{cos^{-1}x}. 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^{-1}x} with respect to x.

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

^{-1}x}

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

**Solution:**

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

Taking natural logarithms **[ln=log _{e}]** on both sides, we get that

ln y = cos^{-1}x as we know that ln(e^{k}) =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^{-1}x is -1/√(1-x^{2}).

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

Putting the value of y, that is, y=e^{cos^{-1}x}, 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 -e ^{cos-1x} 1/√(1-x^{2}).**

**ALSO READ:**

Differentiate e^{sinx}: The derivative of e^{sinx} is e^{sinx} cosx.

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

Differentiate e^{tanx}: The derivative of e^{tanx} is e^{tanx} sec^{2}x.

Differentiate e^{sin^{-1}x}: The derivative of e to the power sin inverse x is e^{sin-1x} 1/√(1-x^{2}).

## FAQs

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

Answer: The derivative of e to the power cos^{-1}x is equal to -e^{cos-1x} 1/√(1-x^{2}).

### Q2: If y=e^{cos-1x}, then find dy/dx?

Answer: If y=e^{cos-1x}, then dy/dx = -e^{cos-1x} 1/√(1-x^{2}).