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Differentiation rules

Product, quotient and chain rule explained with live examples.

f′(x) = —
Sum and difference
Differentiate term by term.
(u ± v)′ = u′ ± v′
Constant multiple
Leave the constant in front of the derivative.
(c·u)′ = c·u′
Product
Derivative of the first times the second, plus the first times the derivative of the second.
(u·v)′ = u′v + uv′
E.g. (x²·sin x)′ = 2x·sin x + x²·cos x
Quotient
Watch the order in the numerator and the square in the denominator.
(u/v)′ = (u′v − uv′)/v²
Chain rule (composite function)
Derivative of the outer function times the derivative of the inner.
[f(g(x))]′ = f′(g)·g′
E.g. (sin(x²))′ = cos(x²)·2x

How to combine the rules

Complex functions are differentiated step by step – break them into parts and apply the appropriate rule to each. The most common mistake is forgetting the chain rule for a composite function (the inner derivative) or mixing up the order in a quotient. The calculator above differentiates using all of these rules automatically, so you can check your own work.

Basic formulas

The rules combine with the basic derivative formulas for individual functions. Together, they're enough to differentiate practically any expression.

Tip: try typing a product, quotient, or composite function into the field and watch the resulting derivative.