20 min read

Differentiating Polynomials

Practice combining every rule so far on full polynomial expressions.

Putting It All Together

A polynomial is an expression made of power terms added or subtracted together, like 2 x^{3} - 4 x^{2} + 5 x - 7 . Differentiate it one term at a time, using the power rule on each, and remembering that a lone constant always differentiates to 0 .

Example 1 — a full polynomial

Differentiate f(x) = 2 x^{3} - 4 x^{2} + 5 x - 7 . 1 Derivative of 2 x^{3} : 2 \times 3 x^{2} = 6 x^{2} 2 Derivative of 4 x^{2} : 4 \times 2 x = 8 x 3 Derivative of 5 x : 5 4 Derivative of - 7 (a constant): 0 Answer: f'(x) = 6 x^{2} - 8 x + 5

Example 2 — evaluating the derivative at a point

Using f'(x) = 6 x^{2} - 8 x + 5 from Example 1, find f'(2). 1 Substitute x = 2 : f'(2) = 6 (2)² - 8 (2) + 5 2 Work out each part: 6 (4) - 16 + 5 = 24 - 16 + 5 Answer: f'(2) = 13

Practice

f(x) = x^{3} - 3 x^{2} + 2 . Find f'(1).

Your answer

20 min read

Practice combining every rule so far on full polynomial expressions.

A polynomial is an expression made of power terms added or subtracted together, like 2 x^{3} - 4 x^{2} + 5 x - 7 . Differentiate it one term at a time, using the power rule on each, and remembering that a lone constant always differentiates to 0 .

Example 1 — a full polynomial

Differentiate f(x) = 2 x^{3} - 4 x^{2} + 5 x - 7 . 1 Derivative of 2 x^{3} : 2 \times 3 x^{2} = 6 x^{2} 2 Derivative of 4 x^{2} : 4 \times 2 x = 8 x 3 Derivative of 5 x : 5 4 Derivative of - 7 (a constant): 0 Answer: f'(x) = 6 x^{2} - 8 x + 5

Example 2 — evaluating the derivative at a point

Using f'(x) = 6 x^{2} - 8 x + 5 from Example 1, find f'(2). 1 Substitute x = 2 : f'(2) = 6 (2)² - 8 (2) + 5 2 Work out each part: 6 (4) - 16 + 5 = 24 - 16 + 5 Answer: f'(2) = 13

Practice

f(x) = x^{3} - 3 x^{2} + 2 . Find f'(1).

Your answer