The derivative of x^5 (x to the power 5) is equal to 5x^{4}. Here, we will learn how to find the derivative of x^{5} using the power rule and the first principle of derivatives.

The derivative of x^{5} is denoted by d/dx (x^{5}). The derivative formula of x^{5} is as follows:

$\dfrac{d}{dx}(x^5)=5x^4$.

## Derivative of x^5 by Power Rule

We know that the derivative of x^{n} by the power rule is given by the formula:

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

Put n=5.

So the derivative of x^{5} by the power rule will be equal to $\dfrac{d}{dx}(x^5)$ = 5x^{5-1} = 5x^{4}.

**You can Read:** Derivative of x^{n}: Formula, Proof

## Derivative of x^5 by First Principle

The derivative of a function f(x) by the first principle is given by $\dfrac{d}{dx}$(f(x)) = lim_{h→0} $\dfrac{f(x+h)-f(x)}{h}$.

Put f(x)=x^{5}.

So the derivative of x^{5} will be equal to

$\dfrac{d}{dx}(x^5)$ = lim_{h→0} $\dfrac{(x+h)^5-x^5}{h}$

[Let x+h=z, so that z→x when h→0. Note h=z-x]

So, $\dfrac{d}{dx}(x^5)$

= lim_{z→x} $\dfrac{z^5-x^5}{z-x}$

= 5x^{5-1} using the formula: lim_{x→a} $\dfrac{x^n-a^n}{x-a}$ = na^{n-1}.

= 5x^{4}.

So the derivative of x^5 is 5x^{4} and this is obtained by the first principle of derivative.

**Also Read:** Derivative of x^{4} by first principle

Derivative of e^{sinx} by first principle

Derivative of e^{cosx} by first principle

## Question-Answer

As an application, we now find the derivative of cosx to the power 5.

**Question:** What is the derivative of cos^{5}x?

*Answer:*

Let z=cosx. So dz/dx = -sinx

By the chain rule of derivatives,

$\dfrac{d}{dx}(\cos^5 x)=\dfrac{d}{dx}(z^5)$

= $\dfrac{d}{dz}(z^5)$ \cdot \dfrac{dz}{dx}$

= $5z^4 \cdot (-\sin x)$

= – 5cos^{4}x sinx as z=cosx.

So the derivative of cos^5x is equal to 4- 5cos^{4}x sinx.

**ALSO READ:** **Derivative of 1/(1+x)**

**Derivative of e ^{3x}** |

**Derivative of xlog**

^{x}## FAQs

### Q1: What is the derivative of x^{5}?

**Answer:** The derivative of x^{5} is equal to 5x^{4}.