Derivative of x^e (x to the power e)

The derivative of x^e (x to the power e) is equal to exe-1. Here we will learn to differentiate x^e with respect to x. The derivative of xe is denoted by d/dx (xe) or (xe)’. Its formula is given by

$\dfrac{d}{dx}(x^e)=ex^{e-1}$.

Derivative of x^e

Derivative of x to the power e

Question: Find the derivative of xe, that is, Find

$\dfrac{d}{dx}(x^e)$.

Answer:

As the number e is an irrational number, so it is a constant, whose value lies between 2 and 3.

Thus, we can apply the power rule of derivatives: $\dfrac{d}{dx}$(xn) = nxn-1 in order to get the derivative of xe.

Applying this formula, we obtain that

$\dfrac{d}{dx}$(xe) = exe-1.

So the derivative of xe is equal to exe-1, and this is obtained by the power rule of derivatives.

Question-Answer

Question: Find the derivative of $x^{e^2}$ (x to the power e2).

Answer:

As e2 is a constant, by the power rule of derivatives,

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

So the derivative of x to the power e^2 is equal to $e^2 x^{e^2-1}$.

More Derivatives:

What is the Derivative of e1/x?

What is the Derivative of π?

Find the derivative of 2x

Find the derivative of e2

FAQs

Q1: What is the derivative of x^e?

Answer: The derivative of x^e is equal to exe-1.

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