Power Rule in Differentiation (Part 2)

Power Rule in Differentiation (Part 2)

How to solve the power rule in differentiation problems (when the exponent is a non-zero integer): formula, proof, example, and its solution.

Formula

[x^n]' = n*x^(n - 1). n: Non-zero integer

Previously, you've learned that
[xn]' = nxn - 1
when n is a natural number.

Power rule in differentiation (Part 1)

This is also true
when n is a non-zero integer: -1, -2, -3, ... .

Proof

Power Rule in Differentiation (Part 2): Proof of the Formula

Set n = -m.

Then [xn]' = [x-m]'.

Negative exponent

Differentiate 1/xm.

Reciprocal rule in differentiation

Negative exponent

Change -m to n.

Then [xn]' = nxn - 1.

Example

Find the derivative of the given function. y = (6x^6 - 3x^3 + 2x^2 - 1)/x^3

Divide each term on the numerator by x3.

Then y = 6x3 - 3 + 2x-1 - x-3.

Use the power rule to find y'.

y' = 6⋅3x2 - 0 + 2⋅(-1)x-2 - (-3)x-4

Change the negative exponents into the fractions.

Then y' = 18x2 - 2/x2 + 3/x4.