Practice Differential calculus: derivatives
A derivative describes the instantaneous rate of change of a function and the slope of its tangent.
It helps you study motion, growth and local extrema.
Exercises with solutions
The derivative gives the slope of the graph at every point. With the power rule the exponent becomes a factor and decreases by one. Differentiate sums term by term; constant factors stay, and a bare number disappears.
Exercise 1
Differentiate f(x) = x³.
Show solution and working
f′(x) = 3x²
By the power rule the exponent 3 becomes a factor and the new exponent is 3 − 1 = 2.
Exercise 2
Differentiate f(x) = 5x² − 3x + 7.
Show solution and working
f′(x) = 10x − 3
Term by term: 5x² gives 10x, −3x gives −3, and the constant 7 disappears.
Exercise 3
Differentiate f(x) = 4x.
Show solution and working
f′(x) = 4
Since x = x¹, the derivative is 4 · 1 · x⁰ = 4. A straight line has the same slope everywhere.
Exercise 4
What is the slope of f(x) = x² at x = 3?
Show solution and working
6
Differentiate first: f′(x) = 2x. Then substitute: f′(3) = 6.
Common mistake
Carrying constants along. The derivative of a bare number is zero because the function value does not change there. Also check whether the question asks for the slope at a point or for the derivative function.
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