Second and higher derivatives

What you will be able to do

• write and find second and higher derivatives, in the several notations;
• say what the sign of $f''$ tells about the graph;
• find $\dfrac{d^2y}{dx^2}$ for a parametric curve, and verify a relation between $y$ and its derivatives.

The derivative of the derivative

Differentiate $y = f(x)$ once and you get the slope function $f'(x)$. Differentiate again and you get the second derivative. The notations are

The third derivative is $f'''(x)$ or $\dfrac{d^3y}{dx^3}$, and the $n$th derivative is $f^{(n)}(x)$. In $\dfrac{d^2y}{dx^2}$ the small $2$ is an order, not a power.

For example, $y = x^4$ gives $y' = 4x^3$, then $y'' = 12x^2$, then $y''' = 24x$. For $y = \sin 2x$, $y' = 2\cos 2x$ and $y'' = -4\sin 2x$.

What the second derivative says

The first derivative is a slope. So the second derivative is the rate at which the slope changes.

• Where $f''(x) > 0$ the slope is increasing, and the graph bends upwards, like a bowl.
• Where $f''(x) < 0$ the slope is decreasing, and the graph bends downwards, like an arch.
• If $x$ is time and $f$ is position, $f'$ is velocity and $f''$ is acceleration.

The words differ between books. IB and AP say concave up and concave down. A-Level says convex where $f'' > 0$ and concave where $f'' < 0$. Indian texts say concave upward and concave downward. Always check which words a question uses.

The second derivative of a parametric curve

For $x = f(t)$ and $y = g(t)$, we have $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$. Call this slope $p$. It is a function of $t$. To differentiate $p$ with respect to $x$, use the chain rule once more:

Do not differentiate $\dfrac{d^2y/dt^2}{d^2x/dt^2}$. That is a common error.

A worked example

Let $x = t^2$ and $y = t^3$. Find $\dfrac{d^2y}{dx^2}$ in terms of $t$.

1. The first slope. $\dfrac{dx}{dt} = 2t$ and $\dfrac{dy}{dt} = 3t^2$, so $\dfrac{dy}{dx} = \dfrac{3t^2}{2t} = \dfrac{3t}{2}$.
2. Differentiate this with respect to $t$. $\dfrac{d}{dt}\left(\dfrac{3t}{2}\right) = \dfrac{3}{2}$.
3. Divide by $\dfrac{dx}{dt}$. $\dfrac{d^2y}{dx^2} = \dfrac{3/2}{2t} = \dfrac{3}{4t}$.

A relation between a function and its derivatives

Some questions ask you to show that $y$ satisfies an equation with $y'$ and $y''$. Differentiate twice, substitute, and simplify.

Take $y = \sin(\ln x)$ for $x > 0$. Then

So $x^2y_2 + xy_1 + y = -\sin(\ln x) - \cos(\ln x) + \cos(\ln x) + \sin(\ln x) = 0$.

Practise with new numbers

Try it in Python

The program estimates a second derivative from three function values, $\dfrac{f(x + h) - 2f(x) + f(x - h)}{h^2}$, and compares it with the exact value. Then it reads the bend of $x^3 - 3x$ from the sign of $f''$.

Common mistakes

• Reading the order as a power. In $\dfrac{d^2y}{dx^2}$ the $2$ counts how many times you differentiate.
• A wrong parametric formula. Differentiate $\dfrac{dy}{dx}$ with respect to $t$ first, then divide by $\dfrac{dx}{dt}$.
• Mixing $f''$ with $(f')^2$. The second derivative is not the square of the first.
• Confusing the bend with the slope. A graph can be falling and still bend up.

Remember these

• $f''$ is the derivative of $f'$. It measures how fast the slope changes.
• $f'' > 0$: bends up. $f'' < 0$: bends down.
• For a parametric curve, $\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right) \Big/ \dfrac{dx}{dt}$.
• To verify a relation, differentiate twice and substitute.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find $y''(1)$ for $y = 2x^5 - x^3$.

2. Find $\dfrac{d^2y}{dx^2}$ for $y = \sin 2x$.

3. Does the graph of $f(x) = x^3$ bend up or down at $x = 2$? And at $x = -2$?

4. For $x = t^2$ and $y = t^3$, what is $\dfrac{d^2y}{dx^2}$ at $t = 3$?

5. The position of a lift is $s = t^3 - 6t^2$ metres at time $t$ seconds. Find its acceleration at $t = 3$ seconds.

References

• OpenStax, Calculus Volume 1, Chapter 3, section on higher-order derivatives, and Chapter 4, section on derivatives and the shape of a graph.
• NCERT, Mathematics Part I for Class XII, Chapter 5, section on second order derivative.