How to Differentiate a Curve You Cannot Solve for y

Functions such as $\sin (x^2)$, curves such as $x^2+y^2=25$, and paths described in terms of time cannot be differentiated with the power, product and quotient rules alone. The chain rule, implicit differentiation and parametric differentiation extend calculus to all of them.

This lesson covers the chain rule, implicit and parametric differentiation, and differentiating one function with respect to another.

**If $y=f(u)$ and $u=g(x)$, then $\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$ — differentiate the outer function at the inner function, then multiply by the derivative of the inner function.

Worked examples:**

- $\frac{d}{dx}\sin (x^2)=2x\cos (x^2)$
- $\frac{d}{dx}(3x^2+1{)}^5=5(3x^2+1{)}^4\times 6x=30x(3x^2+1{)}^4$
- $\frac{d}{dx}{e}^{\tan x}=e^{\tan x}{\sec }^{2}x$

Longer chains. For $y=\log (\cos x^3)$:

$$\frac{dy}{dx}=\frac{1}{\cos x^3}\times (-\sin x^3)\times 3x^2=-3x^2\tan x^3$$

Numerical check. For $y=(3x^2+1{)}^5$ at $x=1$, the formula gives $30\times 4^4=7680$.

An everyday example. The rate at which a balloon's volume grows with time is the rate of change of volume with radius multiplied by the rate of change of radius with time — a chain rule.

The substance. Forgetting the inner derivative is the commonest error — the derivative of $\sin 3x$ is $3\cos 3x$, not $\cos 3x$.

**For an equation linking x and y that is not solved for y, differentiate every term with respect to x, treating y as a function of x so that each y-term gains a factor $\frac{dy}{dx}$, then collect those terms and solve for $\frac{dy}{dx}$.

Worked example.** Find $\frac{dy}{dx}$ for $x^2+y^2=25$.

$$2x+2y\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{x}{y}$$

At $(3,4)$ the slope is $-\frac{3}{4}$, perpendicular to the radius, whose slope is $\frac{4}{3}$.

Worked example 2. For $x^3+y^3=3xy$, differentiating gives $3x^2+3y^2\frac{dy}{dx}=3y+3x\frac{dy}{dx}$, so

$$\frac{dy}{dx}=\frac{y-x^2}{y^2-x}$$

At $(1.5,1.5)$, which lies on the curve, the slope is $\frac{1.5-2.25}{2.25-1.5}=-1$.

An everyday example. A ladder sliding down a wall keeps $x^2+y^2$ equal to the square of its length, and implicit differentiation links how fast its foot and its top move.

The substance. The product rule still applies to terms like xy — $\frac{d}{dx}(xy)=y+x\frac{dy}{dx}$, not just $\frac{dy}{dx}$.

**If $x=f(t)$ and $y=g(t)$, then $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$, provided $\frac{dx}{dt}\ne 0$.

Worked example.** For the circle $x=5\cos t$, $y=5\sin t$:

$$\frac{dx}{dt}=-5\sin t,\frac{dy}{dt}=5\cos t,\frac{dy}{dx}=-\cot t$$

At $t=\frac{\pi }{4}$ the slope is $-1$.

Worked example 2. For the parabola $x=a{t}^2$, $y=2at$, $\frac{dy}{dx}=\frac{2a}{2at}=\frac{1}{t}$. With $a=2$ and $t=3$, the point is $(18,12)$ and the slope is $\frac{1}{3}$.

Worked example 3. For $x=a(\theta -\sin \theta )$, $y=a(1-\cos \theta )$, $\frac{dy}{dx}=\frac{\sin \theta }{1-\cos \theta }=\cot \frac{\theta }{2}$.

An everyday example. A javelin whose horizontal and vertical positions are each known in terms of time follows a parametric path; dividing the two speeds gives the slope of its flight at any instant.

The substance. The parameter need not be eliminated — finding the Cartesian equation first is usually slower and can hide which part of the curve you are on.

**To differentiate u with respect to v, where both are functions of x, find $\frac{du}{dx}$ and $\frac{dv}{dx}$ and divide: $\frac{du}{dv}=\frac{du/dx}{dv/dx}$.

Worked example.** Differentiate ${\sin }^{2}x$ with respect to $e^{\cos x}$.

- $u={\sin }^{2}x$ gives $\frac{du}{dx}=2\sin x\cos x$
- $v=e^{\cos x}$ gives $\frac{dv}{dx}=-\sin x{e}^{\cos x}$
- $\frac{du}{dv}=\frac{2\sin x\cos x}{-\sin x{e}^{\cos x}}=-2\cos x{e}^{-\cos x}$

Worked example 2. Differentiate ${\tan }^{-1}\frac{2x}{1-x^2}$ with respect to ${\sin }^{-1}\frac{2x}{1+x^2}$ for $|x|<1$. With $x=\tan \theta$, both equal $2\theta$, so the derivative is 1.

An everyday example. Comparing how a car's fuel use changes with its speed, when both are recorded against time, is differentiating one function with respect to another.

The substance. This is parametric differentiation with x as the parameter — the same division of two rates.