Power Rule in Differentiation (Part 1)

Power Rule in Differentiation (Part 1)

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

Formulas

[x^n]' = n*x^(n - 1). [C]' = 0. n: Natural number, C: Constant (that has no variable x)

The derivative of xn is nxn - 1.

This is true when n is a natural number.

(Actually, this is true when n is a real number.
But you'll see the other cases later.)

Power rule in differentiation (Part 2)

Power rule in differentiation (Part 3)

Power rule in differentiation (Part 4)

The derivative of a constant number C is 0.

This is the case when n = 0.
(= the constant has no variable x.)

Proofs

Power Rule in Differentiation (Part 1): Proof of the Formula 1

Let's see the proof of [xn]' = nxn - 1.

Use the definition of a derivative function.

Expand (x + h)n.

Binomial theorem

Simplify the first two terms
using nC0 = 1 and nC1 = n.

And leave the other terms.

Combinations (nCr)

Cancel the dark gray terms.

Divide each term in the numerator by h.

nxn - 1 term doesn't have h.
So, as h → 0, nxn - 1 remains.

The other terms all have h.
So, as h → 0, they all go to 0.

So [xn]' = nxn - 1.

Power Rule in Differentiation (Part 1): Proof of the Formula 2

Let's see the proof of [C]' = 0.

Use the definition of the derivative function.

f(x + h) = f(x) = C.

The numerator is real 0.

So the limit is 0.

So [C]' = 0.

Example 1

For the given f(x), find f'(x). f(x) = x^2

The exponent of x2 is 2.

So f'(x) is,
write the exponent, 2,
write x,
and reduce the exponent to, 2 - 1, 1.

So f'(x) = 2x.

Example 2

For the given f(x), find f'(x). f(x) = x^5

The exponent of x5 is 5.

So f'(x) is,
write the exponent, 5,
write x,
and reduce the exponent to, 5 - 1, 4.

So f'(x) = 5x4.

Example 3

For the given f(x), find f'(x). f(x) = x

The exponent of x is 1.

So f'(x) is,
write the exponent, 1,
write x,
and reduce the exponent to, 1 - 1, 0.

So f'(x) = 1.

Zero exponent

Example 4

For the given f(x), find f'(x). f(x) = 14

f(x) is a constant function:
it has no variable x.

So f'(x) = 0.