Differentiating inverse functions and implicit relationshipsEdexcel International A Level Maths: Revision notes

If $x$ is given as a function of $y$, the gradient $\frac{dy}{dx}$ is the reciprocal of $\frac{dx}{dy}$: $\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}},\qquad \frac{dx}{dy}\neq0.$ The reason is that $\frac{dy}{dx}$ and $\frac{dx}{dy}$ describe the same tangent, once as rise over run and once as run over rise. Use this rule whenever $x=f(y)$ is easy to differentiate but rearranging to $y=\ldots$ is awkward or impossible.

First differentiate $x$ with respect to $y$, using the standard results and the chain rule:

• $x=\sin ky\Rightarrow\frac{dx}{dy}=k\cos ky$
• $x=\cos ky\Rightarrow\frac{dx}{dy}=-k\sin ky$
• $x=\tan ky\Rightarrow\frac{dx}{dy}=k\sec^2ky$
• $x=e^{ky}\Rightarrow\frac{dx}{dy}=ke^{ky}$
• $x=\ln(ay+b)\Rightarrow\frac{dx}{dy}=\frac{a}{ay+b}$ Example: $x=\sin 3y$ gives $\frac{dx}{dy}=3\cos 3y$, so $\frac{dy}{dx}=\frac{1}{3\cos 3y}$.

Often the question asks for $\frac{dy}{dx}$ in terms of $x$. Use an identity to replace the $y$ terms. Example: $x=\tan 2y$. Then $\frac{dx}{dy}=2\sec^22y=2(1+\tan^22y)=2(1+x^2)$, so $\frac{dy}{dx}=\frac{1}{2(1+x^2)}.$ Example: $x=\ln(2y+1)$. Then $\frac{dx}{dy}=\frac{2}{2y+1}$ and $\frac{dy}{dx}=\frac{2y+1}{2}$. Since $e^x=2y+1$, this is also $\frac12e^x$. Check: making $y$ the subject gives $y=\frac{e^x-1}{2}$, which differentiates to $\frac12e^x$.

Find the gradient at a given point by substituting the y-value into $\frac{dx}{dy}$ first, then taking the reciprocal. Example: $x=e^{2y}+y$ at $y=0$. The point is $(1,0)$ and $\frac{dx}{dy}=2e^{0}+1=3$, so $\frac{dy}{dx}=\frac13$. Tangent: $y-0=\frac13(x-1)$. The normal has gradient $-1\div\frac13=-3$: $y=-3(x-1)$. For $x=\sin 3y$ at $y=\frac{\pi}{18}$: $\frac{dy}{dx}=\frac{1}{3\cos\frac{\pi}{6}}=\frac{2\sqrt3}{9}$.

• If $\frac{dx}{dy}=0$ at a point, $\frac{dy}{dx}$ is undefined there: the tangent is vertical and the rule cannot be used.
• A sign check: if $\frac{dx}{dy}>0$ then $\frac{dy}{dx}>0$ too, because a reciprocal keeps its sign.
• Always give exact values (such as $\frac{2\sqrt3}{9}$) when the question says exact.