State the product rule.

\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}

State the quotient rule.

\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}

Differentiate \frac{x}{x^2+1}.

\frac{(x^2+1)-x(2x)}{(x^2+1)^2}=\frac{1-x^2}{(x^2+1)^2}

How are \frac{dy}{dx} and \frac{dx}{dy} related?

\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}

Connected rates: how do you find \frac{dr}{dt} from \frac{dV}{dt} and V(r)?

\frac{dr}{dt}=\frac{dV}{dt}\div\frac{dV}{dr}

In a connected rates question, when do you substitute numbers?

After differentiating, not before.

Differentiate \ln(x^2+1).

\frac{2x}{x^2+1}, by the chain rule

Differentiate \sin^2x.

2\sin x\cos x, by the chain rule

Product, quotient and chain rulesAQA A-Level Maths: Flashcards

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State the chain rule.

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State the chain rule.: $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$
State the product rule.: $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$
State the quotient rule.: $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$
Differentiate $(2x+1)^5$ .: $10(2x+1)^4$
Differentiate $x^2e^{3x}$ .: $(2x+3x^2)e^{3x}$
Differentiate $\frac{x}{x^2+1}$ .: $\frac{(x^2+1)-x(2x)}{(x^2+1)^2}=\frac{1-x^2}{(x^2+1)^2}$
How are $\frac{dy}{dx}$ and $\frac{dx}{dy}$ related?: $\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$
Connected rates: how do you find $\frac{dr}{dt}$ from $\frac{dV}{dt}$ and $V(r)$ ?: $\frac{dr}{dt}=\frac{dV}{dt}\div\frac{dV}{dr}$
What is $\frac{dV}{dr}$ for $V=\frac43\pi r^3$ ?: $4\pi r^2$
In a connected rates question, when do you substitute numbers?: After differentiating, not before.
Differentiate $\ln(x^2+1)$ .: $\frac{2x}{x^2+1}$ , by the chain rule
Differentiate $\sin^2x$ .: $2\sin x\cos x$ , by the chain rule

Exam questions on Product, quotient and chain rules

A curve has equation $y=x^2e^{3x}$ .
Show that the stationary point at $x=0$ is a minimum.2 marks
A curve has equation $y=\frac{3x+1}{x^2+1}$ .
Find the $x$ -coordinates of the stationary points of the curve, correct to 3 significant figures.2 marks
A curve has equation $y=\sqrt{4x^2+9}$ .
Find $\frac{dy}{dx}$ , simplifying your answer.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).