Key takeaways
- A derivative is the slope of a curve at one point, or the instant rate of change.
- Power rule: d/dx (xⁿ) = n·xⁿ⁻¹.
- Product rule: (u·v)′ = u′·v + u·v′. Quotient rule: (u/v)′ = (u′·v − u·v′) ÷ v².
- Chain rule: differentiate the outside, keep the inside, then multiply by the inside's derivative.
- The derivative of a product is NOT the product of the derivatives.
In this guide
What is a derivative?
A derivative tells you how fast a function is changing at one exact point. On a graph, it is the slope of the tangent line: the straight line that just touches the curve at that point.
For a straight line, the slope is the same everywhere. For a curve like y = x², the slope is different at every point. The derivative gives you a formula for that changing slope.
Why do derivatives matter?
Derivatives measure change. Speed is the derivative of position. Acceleration is the derivative of speed. In business, the derivative of cost tells you the cost of making one more item.
They also find the best value. At the top of a hill or the bottom of a valley, the slope is zero. Setting the derivative to zero finds maximum profit, minimum material or the highest point of a throw.
A derivative turns 'how does this change?' into a formula you can calculate.
The rules of differentiation
| Rule | Use it for | Formula | Example |
|---|---|---|---|
| Power | A single power of x | d/dx (xⁿ) = n·xⁿ⁻¹ | d/dx (x³) = 3x² |
| Product | Two functions multiplied | (u·v)′ = u′·v + u·v′ | d/dx (x²·sin x) = 2x·sin x + x²·cos x |
| Quotient | One function divided by another | (u/v)′ = (u′·v − u·v′) ÷ v² | d/dx (x ÷ (x + 1)) = 1 ÷ (x + 1)² |
| Chain | A function inside another | dy/dx = dy/du · du/dx | d/dx (3x + 1)⁵ = 15(3x + 1)⁴ |
Here u and v stand for any two functions of x, and u′, v′ are their derivatives.
How to find a derivative step by step
- 1Look at the structure. Is it a sum, a product, a quotient, or a function inside a function?
- 2Split sums. Differentiate each term on its own:
d/dx (x³ + x²) = 3x² + 2x. - 3Pick the rule for each term from the table.
- 4Apply the rule carefully, writing u, v, u′ and v′ separately if needed.
- 5Simplify the result.
- 6Substitute a value if you need the slope at a point.
The chain rule, in detail
For (3x + 1)⁵, the outside is ( )⁵ and the inside is 3x + 1. Differentiate the outside and keep the inside: 5(3x + 1)⁴. Then multiply by the derivative of the inside, which is 3. The answer is 15(3x + 1)⁴. At x = 0, that is 15 × 1⁴ = 15.
Practise each rule with the power rule calculator, product rule calculator, quotient rule calculator and chain rule calculator. Each one works with a set family of power functions and evaluates the derivative at the x you choose, showing every step.
Common mistakes to avoid
- Forgetting the inside derivative in the chain rule.
d/dx (3x + 1)⁵is not5(3x + 1)⁴. You must multiply by 3. - Swapping the order in the quotient rule. It is
u′·v − u·v′on top. The minus makes order matter. - Forgetting to square the bottom in the quotient rule.
- Negative and fraction powers.
d/dx (x⁻¹) = −x⁻², andd/dx (√x) = d/dx (x^(1/2)) = ½·x^(−1/2).
Tips and tricks
- Constants disappear. The derivative of any plain number is 0.
- Constant multiples stay.
d/dx (5x³) = 5 × 3x² = 15x². - Remember four more:
d/dx (sin x) = cos x,d/dx (cos x) = −sin x,d/dx (eˣ) = eˣandd/dx (ln x) = 1 ÷ x.
Where you'll use it in real life
- Physics: velocity and acceleration from a position formula.
- Business: marginal cost and marginal revenue.
- Engineering: finding the maximum stress or the minimum material.
- Machine learning: training models by following the slope downhill (gradient descent).
Quick summary
A derivative is the slope of a curve at a point. Use the power rule d/dx (xⁿ) = n·xⁿ⁻¹ for single powers, the product and quotient rules when functions are multiplied or divided, and the chain rule for a function inside another. Simplify first, keep track of signs, and check each step with the free calculators linked above.
Frequently asked questions
What is the physical meaning of a derivative?
A derivative is an instant rate of change. For example, velocity is the derivative of position with respect to time: how fast the position is changing at one moment.
How do you apply the chain rule?
Find the outer and inner functions. Differentiate the outer function while keeping the inner one unchanged, then multiply by the derivative of the inner function.
When do I use the product rule?
Use it when two functions of x are multiplied together, like x² times sin x. If one of them is just a number, you do not need it: the number simply stays in front.
What is the derivative of a constant?
Zero. A constant never changes, so its rate of change is 0. Its graph is a flat horizontal line with slope 0.
