The power rule of calculus is as follows:

The derivative of x^{n} is n*x^{(n-1)}

Note: the derivative of a constant is zero.

This method of differentiation is used for simple polynomials or rational expressions that can be simplified into a single polynomial.

It is also essential to remember the rules of exponents:

https://www.onlinemathlearning.com/exponent-rules.html

**Example 1 – find the derivative of f(x)=2x ^{2} + 3x + 4**

Let’s apply the power rule to each term in the polynomial:

x^{2}, n = 2:(2)*2x^{2-1} = 4x^{1} = 4x

3x, n = 1: (3) * x^{1-1} = 3x^{0} = 3 * 1 = 3

4: 0 (4 is a constant, so its derivative is zero)

The derivative of f(x) is f’(x) = 4x+3+0 = 4x+3

**Example 2 – find the derivative of f(x) = **

Let’s simplify this term into a single polynomial.

Let’s apply the power rule to each term in the polynomial:

2x^{2}, n = 2:(2)*2x^{2-1} = 4x^{1} = 4x

3x^{1}, n = 1:(1)*3x^{1-1} = 3x^{0}=3 * 1 = 3

4x^{-2}, n = -2:(-2)*4x^{-2-1} = -8x^{-3}

The derivative of f(x) is f’(x) = 4x+3-8x^{-3}

**Example 3 – find the derivative of f(x)= at x=3**

Let’s simplify this term into a single polynomial.

Let’s apply the power rule to each term in the polynomial:

x, n=1: 1*x^{1-1}=1x^{0}=1*1=1

-3: 0 (since 3 is a constant, the derivative is zero)

The derivative of f(x) is 1 +0 = 1 = f’(x)

Since the derivative is a constant (1), the derivative at any x value is 1.

The graph of the original function proves this:

**Example 4 – find the derivative of f(x)=2x ^{-3} + 4x^{-2} + x^{-1}**

Let’s apply the power rule to each term in the polynomial:

Combining the results, the derivative of f(x) is f’(x) =