Parametric and second derivatives

In some problems $x$ and $y$ are both given as functions of a third variable $t$ , typically time or angle , rather than $y$ being given as a function of $x$. Examples: a particle moving along a curve, or a circle parametrised as $(\cos\theta, \sin\theta)$. The parametric derivative formula expresses $\dfrac{dy}{dx}$ without first eliminating the parameter.

A second derivative $\dfrac{d^2 y}{dx^2}$ is the derivative of $\dfrac{dy}{dx}$, taken with respect to $x$ again. Higher orders follow the same pattern.

Parametric derivative

If $x = f(t)$ and $y = g(t)$ are differentiable functions of $t$, with $f'(t) \neq 0$, then

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}.$

This follows from the chain rule: $\dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot \dfrac{dx}{dt}$.

Worked example

For the cycloid $x = a(t - \sin t), y = a(1 - \cos t)$:

$\dfrac{dx}{dt} = a(1 - \cos t)$, $\dfrac{dy}{dt} = a \sin t$.

$\dfrac{dy}{dx} = \dfrac{a \sin t}{a(1 - \cos t)} = \dfrac{\sin t}{1 - \cos t} = \dfrac{2 \sin(t/2)\cos(t/2)}{2 \sin^2(t/2)} = \cot(t/2)$.

Second derivative in parametric form

$\dfrac{d^2 y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{dy}{dx}\right) = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right) \cdot \dfrac{dt}{dx} = \dfrac{(dy/dx)'_t}{dx/dt}$.

In words: differentiate $\dfrac{dy}{dx}$ as a function of $t$, then divide by $\dfrac{dx}{dt}$.

Higher-order derivatives

$f''(x)$, $f'''(x)$, $f^{(n)}(x)$ , repeated differentiation. Standard formulas:

• $\dfrac{d^n}{dx^n}(x^k) = k(k-1)\cdots(k - n + 1) x^{k - n}$ for $k \ge n$.
• $\dfrac{d^n}{dx^n}(e^{ax}) = a^n e^{ax}$.
• $\dfrac{d^n}{dx^n}(\sin x) = \sin(x + n\pi/2)$.
• $\dfrac{d^n}{dx^n}(\cos x) = \cos(x + n\pi/2)$.
• $\dfrac{d^n}{dx^n}(\ln x) = \dfrac{(-1)^{n-1}(n-1)!}{x^n}$.

Worked examples

Example 1. If $x = a\cos\theta, y = a\sin\theta$, find $\dfrac{dy}{dx}$ and $\dfrac{d^2 y}{dx^2}$.

$\dfrac{dx}{d\theta} = -a\sin\theta$, $\dfrac{dy}{d\theta} = a\cos\theta$. So $\dfrac{dy}{dx} = -\cot\theta$.

$\dfrac{d}{d\theta}(-\cot\theta) = \csc^2\theta$. Divide by $\dfrac{dx}{d\theta} = -a\sin\theta$: $\dfrac{d^2 y}{dx^2} = -\dfrac{\csc^2\theta}{a\sin\theta} = -\dfrac{1}{a \sin^3\theta}$.

Example 2. If $x = t^2, y = t^3$, find $\dfrac{dy}{dx}$ and $\dfrac{d^2 y}{dx^2}$.

$\dfrac{dx}{dt} = 2t$, $\dfrac{dy}{dt} = 3t^2$. So $\dfrac{dy}{dx} = \dfrac{3t}{2}$.

$\dfrac{d}{dt}(3t/2) = 3/2$. Divide by $2t$: $\dfrac{d^2 y}{dx^2} = \dfrac{3}{4t}$.

Example 3. If $y = a\sin x + b\cos x$, show $y'' + y = 0$.

$y' = a\cos x - b\sin x$. $y'' = -a\sin x - b\cos x = -y$. So $y'' + y = 0$.

Example 4. Find the $n$-th derivative of $f(x) = \sin x$.

$f^{(n)}(x) = \sin(x + n\pi/2)$.

Example 5. If $y = e^{a \sin^{-1} x}$, show $(1 - x^2) y'' - x y' - a^2 y = 0$.

$y' = e^{a \sin^{-1} x} \cdot \dfrac{a}{\sqrt{1 - x^2}} = \dfrac{a y}{\sqrt{1 - x^2}}$. So $y' \sqrt{1 - x^2} = ay$, square: $(1 - x^2)(y')^2 = a^2 y^2$.

Differentiate: $-2 x (y')^2 + 2(1 - x^2) y' y'' = 2 a^2 y y'$. Divide by $2 y'$ (assuming nonzero): $-x y' + (1 - x^2) y'' = a^2 y$, i.e., $(1 - x^2) y'' - x y' - a^2 y = 0$. $\blacksquare$

Example 6. Find $\dfrac{d^2 y}{dx^2}$ for $y = x \log x$.

$y' = \log x + 1$. $y'' = 1/x$.

Try it yourself

1. If $x = at^2, y = 2at$, find $\dfrac{dy}{dx}$.
2. If $x = 3 \cos t, y = 4 \sin t$, find $\dfrac{dy}{dx}$ and $\dfrac{d^2 y}{dx^2}$.
3. Find the $n$-th derivative of $\cos x$.
4. Find the $n$-th derivative of $e^{ax}$.
5. If $y = \tan^{-1} x$, find $y''$.
6. If $y = a\cos(\log x) + b\sin(\log x)$, show $x^2 y'' + x y' + y = 0$.
7. If $y = \sin^{-1} x$, find $y''$ and show $(1 - x^2) y'' - x y' = 0$.
8. Find $\dfrac{d^2 y}{dx^2}$ for $y = \ln(\sin x)$.
9. If $x = a(\theta - \sin\theta), y = a(1 - \cos\theta)$ (the cycloid), compute $\dfrac{d^2 y}{dx^2}$ at $\theta = \pi/2$.
10. If $y = e^x \cos x$, show $y'' - 2 y' + 2 y = 0$.
11. Find the $5$-th derivative of $\ln x$.
12. If $x = \sec\theta, y = \tan\theta$, find $\dfrac{dy}{dx}$.
13. If $y = x^x$, find $y''$ at $x = 1$.
14. Find $\dfrac{d^2 y}{dx^2}$ for $x = a \cos^3 t, y = a \sin^3 t$ (the astroid).

Pitfalls / Tricks

• For parametric $\dfrac{d^2 y}{dx^2}$, don't divide the second derivatives directly. Differentiate $\dfrac{dy}{dx}$ with respect to $t$ first, then divide by $\dfrac{dx}{dt}$.
• The $n$-th derivative of sin and cos shifts by $n\pi/2$ , a clean formula.
• For functions involving $\sin^{-1}$ or $\cos^{-1}$, deriving a differential equation that the function satisfies is a common JEE technique.

Next, the existence theorems for derivatives.