Differentiation of e^cotx | Differentiate e^cotx

The differentiation of e^cotx is equal to -cosec²x e^cotx. In this post, we will learn how to differentiate e to the power cotx with respect to x.

The following two formulas will be used to find the derivative of e^cotx.

1. d/dx(cotx) = -cosec²x.
2. log_a a^k =k.

Now, we will learn to find the derivative of e to the power cotx with respect to x.

How to Find the derivative of e^cotx

Question: How to differentiate e^cotx?

Solution:

Let y= e^cotx

Taking natural logarithms on both sides, we get that (natural logarithm means the logarithm with base e, i.e. log_e)

log y = cotx

[Here we have used the above formula 2]

Now, differentiate both sides with respect to x. This will give us

$\dfrac{1}{y} \dfrac{dy}{dx}$ = -cosec²x, by the above formula 1.

⇒ $\dfrac{dy}{dx}$ = -y cosec²x

Now put back the value of y, that is, y=e^cotx. Therefore, we obtain that

$\dfrac{d}{dx}$(e^cotx) = – e^cotx cosec²x.

Thus, the differentiation of e^cotx with respect to x is equal to -e^cotx cosec²x.

ALSO READ:

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

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.

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