The derivative of x^6 (x to the power 6) is equal to 6x^{5}. Here, we will find the derivative of x^{6} using the power rule and the first principle of derivatives.

The derivative of x^{6} is denoted by d/dx (x^{6}). The derivative formula of x^{6} is as follows:

$\dfrac{d}{dx}(x^6)=6x^5$.

## Derivative of x^6 by Power Rule

By the power rule of derivatives $\dfrac{d}{dx}(x^n)$ = nx^{n-1}, we get that

$\dfrac{d}{dx}(x^6)$ = 6x^{6-1} = 6x^{5}.

Hence the derivative of x^{6} by the power rule is 6x^{5}.

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

Derivative of 10^{x} | Derivative of x^{10}

## Derivative of x^6 by First Principle

In first principle of derivatives formula $\dfrac{d}{dx}$(f(x)) = lim_{h→0} $\dfrac{f(x+h)-f(x)}{h}$, we put

f(x)=x^{6}.

So the derivative of x^{6} by first principle is given by

$\dfrac{d}{dx}(x^6)$ = lim_{h→0} $\dfrac{(x+h)^6-x^6}{h}$

= lim_{h→0} $\dfrac{(x+h)^6-x^6}{(x+h)-x}$

[Let x+h=z, so that z→x when h→0]

Thus, $\dfrac{d}{dx}(x^6)$

= lim_{z→x} $\dfrac{z^6-x^6}{z-x}$

= 6x^{6-1} obtained by the formula lim_{x→a} $\dfrac{x^n-a^n}{x-a}$ = na^{n-1}.

= 6x^{5}.

So the derivative of x^6 is 6x^{5} and this is proved by the first principle.

**Also Read:** Derivative of 2^{x} by first principle

Derivative of x^{4} by first principle

Derivative of e^{sinx} by first principle

## FAQs

### Q1: What is the derivative of x^{6}?

**Answer:** As d/dx (x^{n}) = nx^{n-1}, the derivative of x^{6} is equal to 6x^{5}.