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Learn Derivatives

Master the rules for slopes and rates of change, one rule at a time.

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A derivative measures how fast a function is changing at a single point, which is the slope of its tangent line there. If position is a function of time, its derivative is velocity. Students first define it with a limit, f'(x) as the limit of (f(x + h) - f(x))/h as h goes to 0, then learn shortcut rules: power, product, quotient and chain. With those, plus the derivatives of sine, cosine, e to the x and ln x, you can differentiate almost any formula you meet in a first course. Learnr builds a focused derivatives course and teaches each rule in short interactive lessons with instant feedback on every step.

How to learn Derivatives, step by step

Compute a few derivatives from the definition. Use the limit of (f(x + h) - f(x))/h on x squared to get 2x. Doing it once by hand shows exactly what every shortcut rule is summarizing.
Drill the power rule until it is automatic. Rewrite roots and fractions as powers first: the square root of x is x to the 1/2, and 1/x squared is x to the -2. Then the derivative of x to the n is n times x to the n minus 1.
Add product and quotient rules. The derivative of x squared times sin x is 2x sin x plus x squared cos x. For quotients, remember low times d-high minus high times d-low, all over low squared, and keep the order.
Spend extra time on the chain rule. For sin(x squared), differentiate the outside and keep the inside, then multiply by the inside's derivative: 2x cos(x squared). Practice naming the inner function out loud before you write anything.
Connect derivatives to graphs and motion. Sketch f and f' side by side, noting where f' is zero, positive and negative. Then locate maxima and minima by setting the derivative equal to zero and testing each critical point.
Practice mixed sets in short daily bursts. Mixed problems force you to choose the rule, which is the real skill on tests. A ten-minute Learnr session each day, kept alive with a streak, builds that recognition quickly.

What you'll learn

Compute a derivative from the limit definition
Apply the power, product, quotient and chain rules accurately
Differentiate trigonometric, exponential and logarithmic functions
Use implicit differentiation and find tangent lines
Interpret derivatives as rates of change and use them to find maxima and minima

What's inside your Derivatives course

A sample path Learnr builds for Derivatives. Your real course adapts to your goal and level.

1. Slope, tangent lines and the limit definition
2. Power rule and linearity
3. Product and quotient rules
4. The chain rule
5. Trig, exponential, log and implicit derivatives
6. Applications: rates, extrema and related rates

How long does it take to learn Derivatives?

With solid algebra, the core rules take about 3 to 5 weeks at 3 to 5 hours a week, and applications like optimization and related rates add another 3 to 4 weeks. Weak algebra or trig skills are the main thing that slows learners down.

What you need before you start

Algebra, especially exponents, fractions and function notation, plus basic trigonometry. A first look at limits helps, since the definition of the derivative is a limit.

Common mistakes to avoid

Forgetting the chain rule, so the derivative of (3x + 1) to the fifth comes out as 5(3x + 1) to the fourth instead of 15(3x + 1) to the fourth.
Treating the derivative of a product as the product of the derivatives. Use the product rule: f'g plus fg'.
Subtracting in the wrong order in the quotient rule, which flips the sign of the answer. Keep the bottom function's term first.

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Derivatives questions, answered

What does a derivative actually tell you?

It tells you the instantaneous rate of change of a quantity, such as speed from position or marginal cost from total cost. On a graph, it is the slope of the tangent line at that point.

Why is the derivative of a constant zero?

A constant function never changes, so its graph is a flat horizontal line with slope zero at every point.

Should I learn limits before derivatives?

A basic understanding of limits helps, because the derivative is defined as a limit. You do not need every limit technique to start using the derivative rules.