The second derivative
Taking the second derivative means differentiating a function twice or finding the derivative of the first derivative. In fact, if the function allows, we can take the third, fourth, fifth and other higher order derivatives.
Let's have a look at what this means for a simple function.
If we start with a function like:
$y=x^5$y=x5
Then, using the power rule, the first derivative is:
$y'=5x^4$y′=5x4
We could then take the derivative of this function, again using the power rule. We call this the second derivative, and use the notation $y''$y′′:
$y''=20x^3$y′′=20x3
We can keep going and differentiate the function again if we wanted to, adding a prime to the notation each time.
$y'''=60x^2$y′′′=60x2
In this course, it is only useful for us to consider up to the second derivative. We will use it to determine the behaviour of functions through identifying key features for graphing and problem solving.
As with functions and the first derivative, there are multiple notations that can be used for the second derivative. You should always be consistent with the notation used within a problem. The table below summarises the different common notations:
| Function | 1st Derivative | 2nd Derivative | 3rd Derivative |
|---|---|---|---|
| $y$y | $y'$y′ | $y''$y′′ | $y'''$y′′′ |
| $f\left(x\right)$f(x) | $f'\left(x\right)$f′(x) | $f''\left(x\right)$f′′(x) | $f'''\left(x\right)$f′′′(x) |
| $y$y | $\frac{dy}{dx}$dydx | $\frac{d^2y}{dx^2}$d2ydx2 | $\frac{d^3y}{dx^3}$d3ydx3 |
For high order derivatives in the notation using primes, the order of the derivative may be superscripted instead such as $f''''\left(x\right)=f^{(4)}\left(x\right)$f′′′′(x)=f(4)(x).
The notation $\frac{d^2y}{dx^2}$d2ydx2 is interpreted as: take the derivative of $y$y twice with respect to $x$x both times, i.e. $\left(\frac{d}{dx}\right)^2y$(ddx)2y.
Just as when finding the first derivative identify the types of functions in the problem - powers, polynomials, exponential and trigonometric, and then look for structure in the function that will dictate what rules need to be applied - such as product, quotient or composites of functions.
Practice questions
Question 1
Suppose $y=\frac{2}{x^5}$y=2x5.
Find $\frac{dy}{dx}$dydx.
Find $\frac{d^2y}{dx^2}$d2ydx2.
Find $\frac{d^3y}{dx^3}$d3ydx3.
Question 2
Find the second derivative of $y=\frac{9}{1+x}$y=91+x.