Derivative : Power Rule

Introduction

In calculus, one of the most important and commonly used techniques for finding derivatives is the Power Rule.
It allows us to quickly find the derivative of functions where the variable $x$ is raised to a power — such as $$x^2 , x^{-5} , x^{\frac{1}{2}} $$

Understanding this rule is essential, because it forms the foundation for differentiating many more complex functions later on.

What is a Derivative?

A derivative measures how a function changes as its input changes.
In simple terms, it tells us the slope of the function’s curve at any given point.

Mathematically, the derivative of a function f(x) is defined as:

$$f^\prime(x)=\lim_{h\to{0}}\frac{f(x+h)-f(x)}{h}$$​

While this definition is important for understanding the concept, in practice we use shortcut rules — such as the Power Rule — to find derivatives quickly.

The Power Rule

If $$f(x)=x^n$$, where n is any real number, then the derivative is given by:

$$f^\prime(x)=\frac{d}{dx} f(x) = n\cdot x^{n-1}$$

This means:

• Bring the exponent n down to the front as a coefficient.

• Subtract 1 from the exponent.

The Power Rule with coefficient

If the function includes a constant multiplier, the constant remains as it is.

$$f(x)=a\cdot x^n$$ then $$f^\prime(x)=\frac{d}{dx} f(x)=a\cdot n x^{n-1}$$

The Addition Rule

if the function is polynomial like

$$f(x)=ax^{n}+bx^{m}+C$$ then derivative of polynomial is

$$f^\prime{x}=a\cdot{n}x^{n-1}+b\cdot{m}x^{m-1}$$

Derivative of rational and irrational functions

if we have rational or irrational functions,

first of all, we need to change these functions exponential form

$$f(x)=\frac{1}{x}=x^{-1}$$ and $$g(x)=\sqrt{x}=x^{\frac{1}{2}}$$

and can find out derivative use power rule

$$f^\prime(x)=-1\cdot{x^{-1-1}}=-x^{-2}$$ and

$$g^\prime(x)=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$$