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:

power rule

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

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)*2x2-1 = 4x1 = 4x
3x, n = 1: (3) * x1-1 = 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) = power rule 1

Let’s simplify this term into a single polynomial.

power rule 2

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

2x2, n = 2:(2)*2x2-1 = 4x1 = 4x
3x1, n = 1:(1)*3x1-1 = 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)= power rule 3 at x=3

Let’s simplify this term into a single polynomial.

power rule 4

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

x, n=1: 1*x1-1=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:

power rule 1

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:

power rule 5

Combining the results, the derivative of f(x) is f’(x) = power rule 6