Linearity, product and quotient

To differentiate sums, products and quotients of known functions, three rules suffice. Beware: the derivative of the product is not the product of the derivatives.

Remark — Linearity and the product rule

• The derivative of the sum is the sum of the derivatives: $(f+g)' = f'+g'$.
• The derivative of the product $f\cdot g$ is not $f'\cdot g'$: it is $(f\cdot g)' = f'\,g + f\,g'$.
• The derivative of the quotient: $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$.

Example — Product and quotient

$(x\sin x)' = x'\sin x + x\cdot(\sin x)' = 1\cdot\sin x + x\cos x = \sin x + x\cos x.$ $\left(\dfrac{x}{\sin x}\right)' = \dfrac{x'\sin x - x(\sin x)'}{\sin^2 x} = \dfrac{\sin x - x\cos x}{\sin^2 x}.$

Links

Topics: Derivatives
Concepts: Derivative · Differentiation rules
Skills: Differentiating · Using formulae