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Exercises on derivatives differentiation rules

Learn to differentiate using the basic rules: power, sum, product, and chain rule. Theory and step-by-step solved derivative exercises, with practice corrected instantly by AI.

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Derivatives: quick theory

The derivative f′(x) measures how fast a function changes. Geometrically, it is the slope of the tangent line at each point.

Basic rules: the derivative of xn is n·xn−1 (power rule); the derivative of a sum is the sum of the derivatives; a multiplying constant is kept.

Product: (f·g)′ = f′·g + f·g′. Chain rule (composite function): you differentiate the outer function and multiply by the derivative of the inner function.

Step-by-step solved exercises on derivatives

Solved examples to show you the method. Then, practice with instant correction.

Exercise 1Differentiate f(x) = x³ − 5x² + 2x.

Step-by-step solution

  1. We differentiate term by term using the power rule.
  2. (x³)′ = 3x²; (−5x²)′ = −10x; (2x)′ = 2.
Result: f′(x) = 3x² − 10x + 2.

Exercise 2Differentiate f(x) = x² · sin x (product rule).

Step-by-step solution

  1. We identify f = x² (f′ = 2x) and g = sin x (g′ = cos x).
  2. We apply (f·g)′ = f′·g + f·g′.
  3. We substitute: 2x · sin x + x² · cos x.
Result: f′(x) = 2x·sin x + x²·cos x.

Exercise 3Differentiate f(x) = (2x + 1)³ (chain rule).

Step-by-step solution

  1. Derivative of the outside (power): 3(2x + 1)².
  2. By the derivative of the inside: (2x + 1)′ = 2.
  3. We multiply: 3(2x + 1)² · 2.
Result: f′(x) = 6(2x + 1)².

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This topic is studied in:

1st year of Bachillerato2nd year of Bachillerato

Frequently asked questions about derivatives

What are the basic differentiation rules?

The power rule (the derivative of xⁿ is n·xⁿ⁻¹), the derivative of a sum is the sum of derivatives, the product rule (f·g)′ = f′·g + f·g′, and the chain rule for composite functions.

What is the chain rule?

It is used to differentiate composite functions: you differentiate the outer function and multiply it by the derivative of the inner function. For example, in (2x+1)³ you differentiate the power and multiply it by the derivative of 2x+1.

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