Math · Calculus · Worked example

Differentiate x²(x+1)

Find the derivative in product form and check by expansion.

Name the factors

Use f=x² and g=x+1.

f^{'} = 2 x, g^{'} = 1

Differentiate each factor once, keeping its partner.

{(f g)}^{'} = 2 x (x + 1) + x^{2}

Expanding the original gives x³+x², whose derivative agrees.

2 x (x + 1) + x^{2} = 3 x^{2} + 2 x

The derivative is 3x²+2x.

Differentiate x(x²+2) in two ways.

Show the answer and explanation

The product rule gives (x²+2)+x(2x). Expanding first gives x³+2x.

Change an input and predict the result before checking it.

Sources and scope

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