Derivative rules: power and chain: printable worksheets

Differentiate polynomials, powers, radicals and compositions with the power and chain rules.

Sample worksheet

The first 6 problems of a real worksheet, made the way the builder makes it. Open it in the builder to change the number of problems, the difficulty and the layout, then print.

f (x) = x^{4} + x^{3}

f (x) = 6 x^{5} + 8 x^{4} + 5 x^{3}

f (x) = - 3 ln (x^{2} + 2)

f (x) = e^{- 9 x^{2} - 8}

f (x) = 7 x^{3} - 6 x^{2}

f (x) = - 9 x^{6} - 9 x^{4} + 7 x^{2}

Show answers

f^{'} (x) = 4 x^{3} + 3 x^{2}

f^{'} (x) = 30 x^{4} + 32 x^{3} + 15 x^{2}

f^{'} (x) = - \frac{6 x}{x ^{2} + 2}

f^{'} (x) = - 18 x e^{- 9 x^{2} - 8}

f^{'} (x) = 21 x^{2} - 12 x

f^{'} (x) = - 54 x^{5} - 36 x^{3} + 14 x

The power rule and the chain rule are the two workhorses of first-semester calculus. The power rule handles terms like a variable raised to a number, including negative exponents and roots once they are rewritten as powers. The chain rule handles a function inside another function, and it is the place where students most often lose a factor. Once both rules feel automatic, the rest of differentiation becomes much more manageable.

This worksheet generates derivative problems for both rules. At the easiest setting it asks only for power rule problems with small whole-number exponents. At the medium setting it adds compositions, where an outer function wraps an inner expression, and at the hardest setting it allows deeper nesting and quotients that are simpler to rewrite before differentiating. You can choose which outer functions appear, such as powers, sine, cosine, the exponential function and the natural logarithm, and which exponents are allowed, including negative values and square or cube roots. Answers can be shown expanded or in factored form, and roots can be written as radicals or as exponents.

Practice suggestions

The most common slip is forgetting to multiply by the derivative of the inside function. Ask students to name the outer and inner functions out loud before they start. The worked steps in the answer key help them find exactly where a mistake began.

Start with the power rule alone until negative and fractional exponents feel comfortable.
Add the chain rule with a single layer, then allow two or three layers.
Rewrite radicals as powers first, then differentiate.

The button above opens this sample in the builder, where you can change the settings and print a new sheet.

What students practise

I can find derivatives with the power and chain rules.

Students differentiate polynomials, powers, radicals and compositions with the power and chain rules and simplify.

Key words

derivative: the rate of change of a function
power rule: bring the exponent down and lower it by one
chain rule: derivative of the outside times derivative of the inside

Common mistakes

Forgets to multiply by the derivative of the inside.
Differentiates a radical without rewriting it as a power.

Before this: Factoring trinomials (Algebra 1)