Derivative Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Derivative.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.
How fast is the output changing right now? The slope of the curve at each point.
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: The derivative transforms a function into its 'slope function'—the rate of change at every input.
Common stuck point: Derivative of position is velocity. Derivative of velocity is acceleration.
Sense of Study hint: Try sketching the tangent line at the point and estimate its slope by picking two close points on the curve.
Worked Examples
Example 1
easySolution
- 1 Apply the power rule to each term: \frac{d}{dx}[x^n] = nx^{n-1}.
- 2 For 3x^2: 3 \cdot 2x^{2-1} = 6x.
- 3 For 5x: 5 \cdot 1x^{1-1} = 5.
- 4 For -2: the derivative of a constant is 0.
- 5 Combine: f'(x) = 6x + 5.
Answer
Example 2
mediumExample 3
hardPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
hardRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.