The derivative of x^10 (x to the power 10) is 10x^{9}. In this post, we will find the derivative of x^{10} by the first principle and the power rule of derivatives.

The derivative of x^{10} is denoted by d/dx (x^{10}), and its formula is given by

$\dfrac{d}{dx}(x^{10})=10x^9$.

## Derivative of x^10 by First Principle

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

Let us Put f(x)=x^{10}.

Hence the derivative of x^{10} using the first principle is equal to

$\dfrac{d}{dx}(x^{10})$ = lim_{h→0} $\dfrac{(x+h)^{10}-x^{10}}{h}$

= lim_{h→0} $\dfrac{(x+h)^{10}-x^{10}}{(x+h)-x}$

[Let x+h=t, so that t→x when h→0]

Thus, $\dfrac{d}{dx}(x^{10})$

= lim_{t→x} $\dfrac{t^{10}-x^{10}}{t-x}$

= 10x^{10-1} by the limit formula: lim_{x→a} $\dfrac{x^n-a^n}{x-a}$ = na^{n-1}.

= 10x^{9}.

So the derivative of x^10 by the first principle is equal to 10x^{9}.

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

Derivative of x^{5} by first principle

Derivative of x^{3/2} | Derivative of 2^{x}

Derivative of e^{sinx} by first principle

Derivative of e^{cosx} by first principle

## Derivative of x^10 by Power Rule

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

n=10.

Thus, $\dfrac{d}{dx}(x^{10})$ = 10x^{10-1} = 10x^{9}.

So the derivative of x^10 by the power rule is 10x^{9}.

**Related Topics:** Derivative of x^{n}: Formula, Proof

## Question-Answer

We now find the derivative of e to the power 10x.

**Question:** Find the derivative of e^{10x}.

*Answer:*

Let z=e^{x}. So dz/dx = e^{x}.

By the chain rule of derivatives,

$\dfrac{d}{dx}(e^{10x})=\dfrac{d}{dx}(z^{10})$

= $\dfrac{d}{dz}(z^{10}) \cdot \dfrac{dz}{dx}$

= $10z^{10-1} \cdot e^x$

= 10 e^{9x} ⋅ e^{x} as z=e^{x}

= 10 e^{10x}.

So the derivative of e^{10x} is equal to 10 e^{10x}.

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

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

**Derivative of xe**

^{-x}## FAQs

### Q1: What is the derivative of x^{10}?

**Answer:** The derivative of x^{10} is equal to 10x^{9}.