 Derivatives of Polynomials How to find the derivative of a polynomial: formulas, proofs, examples, and their solutions.

Formula: Constant Multiple Rule in Differentiation The coefficient k is not affected
by the differentiation.

So [kf(x)]' = kf'(x).

This formula is needed
to solve the derivatives of polynomials.

Formula: Sum Rule (and Subtraction Rule) in Differentiation The plus minus signs (±) are not affected
by the differentiation.

So [f(x) ± g(x)]' = f'(x) ± g'(x).

This formula is also needed
to solve the derivatives of polynomials.

Proofs Take the coefficient k out by factoring.

Take the coefficient k out from the limit.

The limit part is f'(x).

So [kf(x)]' = kf'(x). Rewrite the terms like this:
f(x + h) ± g(x + h) - f(x) ± [-g(x)].

Combine the gray terms:
[f(x + h) - f(x)].

And combine the dark gray terms:
±[g(x + h) - g(x)].

Split the fraction into two parts.

Then the limit of the former fraction is f'(x).
And the limit of the latter fraction is g'(x).

So [f(x) ± g(x)]' = f'(x) ± g'(x).

Example 1 As you've seen above,
the coefficients and the plus minus signs
are not affected by the differentiation.

So focus on the variable parts
and differentiate each term.

[x2]' = 2x1
[x]' = 1x0
[+1]' = 0

Power rule in differentiation (Part 1)

So f'(x) = 2x - 3.

Example 2 Focus on the variable parts
and differentiate each term.

[x5]' = 5x4
[x4]' = 4x3
[x2]' = 2x1
[x]' = 1x0
[+14]' = 0

Power rule in differentiation (Part 1)

So f'(x) = 5x4 - 32x3 + 14x - 3.