How to Find a Derivative
A derivative measures how quickly a function changes. Most derivatives you meet in a first course need just four rules, and this guide teaches each with a short example.
What a derivative means
At any point on a curve, the derivative is the slope of the tangent line. If the function describes distance travelled, its derivative is speed. The notation f′(x) and dy/dx mean the same thing.
The power rule
To differentiate xⁿ, bring the power down as a multiplier and reduce the power by one. Constants have a derivative of zero, and a number multiplying a term stays in place.
Product, quotient and chain rules
- Product rule: differentiate one function at a time, keeping the other: (fg)′ = f′g + fg′.
- Quotient rule: (f/g)′ = (f′g − fg′) / g².
- Chain rule: differentiate the outer function, then multiply by the derivative of the inner function.
Quick reference
| Function | Derivative |
|---|---|
| xⁿ | n·xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
Check your own answers with the calculus solver, and learn the reverse process with the integral solver.
Worked examples
Example 1: Power rule: d/dx (x³ + 2x² − 5)
- x³ becomes 3x².
- 2x² becomes 4x.
- The constant −5 becomes 0.
Final answer: 3x² + 4x
Example 2: Product rule: d/dx (x·ln x)
- Let f = x and g = ln x, so f′ = 1 and g′ = 1/x.
- f′g + fg′ = ln x + x·(1/x) = ln x + 1.
Final answer: ln x + 1
Example 3: Quotient rule: d/dx (x / (x + 1))
- f = x, g = x + 1, so f′ = 1 and g′ = 1.
- Numerator: 1·(x + 1) − x·1 = 1.
- Denominator: (x + 1)².
Final answer: 1 / (x + 1)²
Example 4: Chain rule: d/dx sin(5x)
- Outer function sin has derivative cos.
- Inner function 5x has derivative 5.
Final answer: 5cos(5x)
Common mistakes to avoid
- Applying the product rule as “derivative times derivative”.
- Forgetting to multiply by the inner derivative in a chain rule.
- Reversing the numerator of the quotient rule.
- Reducing the power but forgetting to multiply by the old power.
- Treating a constant as if it had a non-zero slope.
Frequently asked questions
What is the easiest way to learn derivatives?
Practise the power rule until it is automatic, then add one rule at a time.
When do I use the chain rule?
When one function sits inside another, like sin(5x) or (3x + 1)⁴.
What is the derivative of a constant?
Zero, because a constant does not change.
How can I check my derivative?
Enter the function in the calculus solver, or differentiate numerically by testing the slope near a point.
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