Power Rule in Differentiation (Part 4)

Power Rule in Differentiation (Part 4)

How to solve the power rule in differentiation problems (when the exponent is a real number): formula, proofs, examples, and their solutions.

Formula

[x^n]' = n*x^(n - 1). n: Real number

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

Power rule in differentiation (Part 3)

This is also true
when n is a real number.

Proof

Power Rule in Differentiation (Part 4): Proof of the Formula

Set y = xn
when n is a real number.

Then ln both sides,
covering the bases with the absolute value signs.
(The absolute value signs are added
becasue both sides can be (-).)

Natural logarithms

Logarithms of powers

Differentiate both sides.

Implicit differentiation

Derivative of ln x

Multiply y on both sides.

Change y into xn.

Negative power

So [xn]' = nxn - 1
when n is a rational number.

Example 1

Find the derivative of the given function. y = x^sqrt(2)

n = √2

So y' = √2x2 - 1.

Example 2

Find the derivative of the given function. y = x^e

n = e

So y' = exe - 1.