Differentiation Rules Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Differentiation Rules.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A set of standard formulas for finding derivatives of common function types without using the limit definition each time.

Shortcuts so you don't have to use the limit definition every time.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Know the rules: power, product, quotient, chain. Apply them systematically.

Common stuck point: Product rule: (fg)' = f'g + fg', NOT (fg)' = f'g' — you cannot just multiply the individual derivatives.

Sense of Study hint: Write out which rule applies to each piece of the expression before computing anything.

Worked Examples

Example 1

easy
Use the product rule to differentiate f(x) = x^3 \cdot \sin x.

Solution

  1. 1
    Identify the two factors: u = x^3 and v = \sin x.
  2. 2
    Find their derivatives: u' = 3x^2 and v' = \cos x.
  3. 3
    Apply the product rule (uv)' = u'v + uv'.
  4. 4
    Result: f'(x) = 3x^2 \sin x + x^3 \cos x.

Answer

f'(x) = 3x^2 \sin x + x^3 \cos x
The product rule states (fg)' = f'g + fg'. Each factor is differentiated once while the other is kept intact, and the two results are added. Never multiply the individual derivatives together.

Example 2

medium
Use the quotient rule to differentiate f(x) = \dfrac{x^2 + 1}{x - 3}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Differentiate f(x) = x^2 \cos x using the product rule.

Example 2

hard
Differentiate f(x) = \dfrac{e^x}{x^2 + 1} using the quotient rule.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

derivative