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=log_e] on both sides, we get that

ln y = cos^-1x 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^-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 -e^cos-1x 1/√(1-x²).

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 sec²x.

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

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