Mathematics·Derivative·grades 10–11

Find the derivative of the function f(x) = 3x³ − 5x² + 7

Problem

The function f(x) = 3x³ − 5x² + 7 is given. Find its derivative f′(x).

Answer

f′(x) = 9x² − 10x

Step-by-step solution

Step 1. Differentiate term by term
The derivative of a sum is the sum of the derivatives, and a constant factor can be pulled out in front: $(c\cdot u)' = c\cdot u'$ .
Step 2. The first term
$(3x^3)' = 3\cdot 3x^{2} = 9x^2$ by the power rule $(x^n)' = n x^{n-1}$ .

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Step 3. The second and third terms
Step 4. Answer

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Common mistakes

Leaving the +7 in the answer — the derivative of the constant term is zero.
Forgetting to lower the exponent by one.

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