What is \frac{d}{dx}(xy)?

y+x\frac{dy}{dx} (product rule)

What does implicit differentiation mean?

Differentiating both sides of an equation with respect to x, using the chain rule for every y term.

Formula for \frac{dy}{dx} from parametric equations?

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

When is the tangent to a parametric curve parallel to the x-axis?

When \frac{dy}{dt}=0 and \frac{dx}{dt}\ne0.

When is the tangent to a parametric curve parallel to the y-axis?

When \frac{dx}{dt}=0 and \frac{dy}{dt}\ne0.

Implicit and parametric differentiationEdexcel International A Level Maths: Flashcards

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What is $\frac{d}{dx}(y^2)$?

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What is $\frac{d}{dx}(y^2)$ ?: $2y\frac{dy}{dx}$
What is $\frac{d}{dx}(xy)$ ?: $y+x\frac{dy}{dx}$ (product rule)
What is $\frac{d}{dx}(\sin y)$ ?: $\cos y\,\frac{dy}{dx}$
What is $\frac{d}{dx}(e^{y})$ ?: $e^{y}\frac{dy}{dx}$
What does implicit differentiation mean?: Differentiating both sides of an equation with respect to $x$ , using the chain rule for every $y$ term.
Formula for $\frac{dy}{dx}$ from parametric equations?: $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$
For $x^2+y^2=25$ , find $\frac{dy}{dx}$ .: $-\frac{x}{y}$
When is the tangent to a parametric curve parallel to the $x$ -axis?: When $\frac{dy}{dt}=0$ and $\frac{dx}{dt}\ne0$ .
When is the tangent to a parametric curve parallel to the $y$ -axis?: When $\frac{dx}{dt}=0$ and $\frac{dy}{dt}\ne0$ .
Equation of the tangent at $(x_1,y_1)$ with gradient $m$ ?: $y-y_1=m(x-x_1)$
Gradient of the normal when the tangent has gradient $m$ ?: $-\frac1m$
$x=3t^2$ , $y=t^3$ : find $\frac{dy}{dx}$ .: $\frac{3t^2}{6t}=\frac t2$

Exam questions on Implicit and parametric differentiation

A curve $C_1$ has equation $x^3+y^3=9$ , and the point $P(1,2)$ lies on $C_1$ .
Find an equation of the tangent to $C_1$ at $P$ , in the form $x+by=c$ .2 marks
A curve $C_2$ has parametric equations $x=t^2+1,\; y=t^3-3t$ , where $t$ is a real parameter.
Find the coordinates of the points on $C_2$ where the tangent is parallel to the $x$ -axis.2 marks
A curve $C_3$ has parametric equations $x=3\sin t,\; y=2\cos 2t$ , for $0\le t<2\pi$ .
Find $\frac{dy}{dx}$ in terms of $t$ , giving your answer in its simplest form.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).