Method · Calculus
How to Differentiate: Step by Step
Differentiation is a decision procedure. Look at the shape of the expression, and the shape tells you which rule to reach for.
Read the shape first
Knowing how to differentiate is mostly knowing which rule the expression in front of you calls for. Before writing anything, ask what the outermost operation is.
| The expression is | Shape | Rule |
|---|---|---|
| a sum or difference | u + v | differentiate term by term |
| a number times something | k · u | the number stays, differentiate u |
| a power of x | xⁿ | power rule |
| one function inside another | f(g(x)) | chain rule |
| two functions multiplied | u · v | product rule |
| one function over another | u / v | quotient rule |
The outermost operation is the one to act on. In x²·sin(3x) the outermost thing is a product, so the product rule comes first — and the chain rule appears later, inside it, when sin(3x) has to be differentiated.
The power rule and constant multiples
For xⁿ, multiply by the exponent and reduce it by one. A constant multiplier simply waits outside.
- 3x⁵ − 4x + 7a sum, so differentiate term by term
- 15x⁴ − 4 + 0power rule on each; the constant differentiates to zero
- 15x⁴ − 4tidy up
The power rule works for every exponent, not only positive whole ones. x⁻² gives −2x⁻³, and √x, written x^½, gives ½x^−½ — which is 1/(2√x).
The chain rule: the one people forget
When one function sits inside another, differentiate the outside as usual and then multiply by the derivative of the inside. That final multiplication is the whole rule, and leaving it out is the most common error in early calculus.
- sin(2x)a sine with 2x inside it
- cos(2x)differentiate the outside, leaving the inside alone
- cos(2x) · 2multiply by the derivative of the inside
- 2cos(2x)tidy up
It nests. For sin(cos(x²)) you work outward to inward, multiplying by each inside derivative in turn — three factors, one for each layer.
The product rule
For two functions multiplied, differentiate each in turn and add the two results: (uv)′ = u′v + uv′.
- u = x², v = sin xname the two factors
- u′ = 2x, v′ = cos xdifferentiate each
- 2x sin x + x² cos xu′v + uv′
The rule is not (uv)′ = u′v′, which is the tempting shortcut and is wrong. Check it on x·x: the correct answer is 2x, while multiplying the derivatives gives 1.
A constant multiplier does not need the product rule. In 3sin x the 3 is not a function of x, so it simply stays where it is.
The quotient rule
For one function over another: (u/v)′ = (u′v − uv′) / v². The order in the numerator matters, because subtraction is not symmetric.
- u = cos x, v = xtop and bottom
- u′ = −sin x, v′ = 1differentiate each
- (−x sin x − cos x) ÷ x²(u′v − uv′) / v²
You can avoid it entirely when the denominator is a simple power: x⁻² is easier to differentiate than 1/x². Rewriting first is often quicker than applying the rule.
Notation, and finishing the job
Two notations for the same operation. f′(x) is compact; dy/dx names both variables and is clearer when it matters what is changing with respect to what — which it does whenever the derivative represents a rate of change of one measured quantity against another.
Whichever you use, simplify the result before stopping. A derivative left as x · (1/x) + ln x is correct and unusable; ln x + 1 is the form you can set equal to zero to find a stationary point.
Putting them together
Real expressions need several rules, applied from the outside inward. The general name for one function wrapped around another is a composite function, and it is the inside function whose derivative the chain rule asks you to multiply by.
- x² · sin(3x)outermost operation is a product
- 2x sin(3x) + x² · [sin(3x)]′product rule first
- [sin(3x)]′ = 3cos(3x)now the chain rule, inside
- 2x sin(3x) + 3x² cos(3x)put it together
Work outward to inward and finish one rule before starting the next. Trying to apply two at once is where long derivatives go wrong.
Questions about how to differentiate
How do I know which rule to use?
Look at the outermost operation. A sum means term by term, a product means the product rule, a quotient means the quotient rule, and a function inside another means the chain rule.
When exactly does the chain rule apply?
Whenever anything other than a bare x sits inside a function or under a power. sin(2x), (x + 1)³ and e^(x²) all need it; sin x does not.
Is (uv)′ the same as u′v′?
No. The product rule is u′v + uv′. Multiplying the derivatives gives the wrong answer — test it on x·x, where the answer is 2x, not 1.
Can I avoid the quotient rule?
Often. If the denominator is a simple power, rewrite it with a negative exponent and use the power rule instead — 1/x² becomes x⁻², which differentiates in one step.