Derivative of x^π (x to the Power Pi)

The derivative of x^π (x to the power pi) is equal to πx^π-1. In this post, we will learn how to differentiate x to the power pi with respect to x.

The derivative of x^π is denoted by d/dx (x^π) or (x^π)’. Its formula is given by

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

Derivative of x to the power pi

Question: Find the derivative of x^π, that is, Find

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

Answer:

Note that the number π is an irrational number, so it is a constant. Its value is approximately equal to $\dfrac{22}{7}$.

So, to find the derivative of x^π, we can apply the power rule of derivatives. The rule says that the derivative of xⁿ is equal to nx^n-1. That is,

$\dfrac{d}{dx}$(xⁿ) = nx^n-1.

Put n=π. Thus, we get that

$\dfrac{d}{dx}$(x^π) = π x^π-1.

So the derivative of x^π (x power π) is equal to π x^π-1, and this is proved by the power rule of derivatives.

Question-Answer

Question: What is the derivative of 2π (2 to the power pi).

Answer:

As both 2 and π are constants, we conclude that 2^π is also a constant.

We know that the derivative of a constant is zero, that is,

$\dfrac{d}{dx}$ (constant) = 0.

So, $\dfrac{d}{dx}(2^\pi)=0$.

Thus, the derivative of 2 to the power pi is equal to 0.

More Derivatives:

What is the Derivative of π?

Find the derivative of 2^x

Find the derivative of e²

Derivative of e^sinx

Derivative of e³

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