Derivative Rules – The Power Rule

The power rule of calculus is as follows:

The derivative of xⁿ 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:

Example 1 – find the derivative of f(x)=2x² + 3x + 4

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

x², n = 2:(2)*2x^2-1 = 4x¹ = 4x
3x, n = 1: (3) * x^1-1 = 3x⁰ = 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:

2x², n = 2:(2)*2x^2-1 = 4x¹ = 4x
3x¹, n = 1:(1)*3x^1-1 = 3x⁰=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=1x⁰=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) =