What Is Derivative?
A derivative measures how quickly a function changes as its input changes. Geometrically, it gives the slope of the tangent line to a curve at a point.
Examples of Derivative
1
If f(x) = x², then f′(x) = 2x.2
If f(x) = 3x + 4, then f′(x) = 3.3
If f(x) = sin x, then f′(x) = cos x.Formula and Key Points
Formula / rule
f′(x) = lim(h→0) [f(x+h) - f(x)] / h
- Derivatives describe rates of change.
- The derivative of a constant is 0.
- The power rule is d/dx(xⁿ) = nxⁿ⁻¹.
- Derivatives are used in motion, optimisation and curve analysis.
Why This Matters in Computer Science
Used in optimization, simulation, graphics, machine learning, control systems and numerical computing.
Worked Example
Problem
Differentiate f(x) = x³ + 2x.
- Differentiate x³ using the power rule: 3x².
- Differentiate 2x: 2.
- Add the derivatives together.
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Final Answerf′(x) = 3x² + 2
Function vs Derivative
| Feature | Function | Derivative |
|---|---|---|
| Meaning | Gives an output value | Gives the rate of change |
| Example | f(x) = x² | f′(x) = 2x |
| Graph idea | Original curve | Slope behaviour of the curve |
Related Math Terms
Derivative: Frequently Asked Questions
What does a derivative represent?
It represents an instantaneous rate of change or the slope of a tangent line.
What is the derivative of x²?
The derivative is 2x.
What is the derivative of a constant?
Zero.