Higher-Order Derivatives (College Board AP® Calculus AB): Revision Note

The second derivative of a function is the derivative, differentiated

• I.e. it is the derivative of the derivative

• It may also be referred to as the second differential

The second derivative may be written as:

• $f^{\text{'}\text{'}}\left(x\right)$

• $\frac{d^{2}y}{d{x}^2}$

• $y^{\text{'}\text{'}}$

The second derivative is the rate of change of the rate of change

• E.g. If a function is increasing (first derivative), then

  • the second derivative describes how rapidly its rate of increase is increasing (or decreasing)

Second derivatives are useful when investigating the shapes of the graphs of functions

• See the study guide on 'Concavity of Functions' for more about this

To find a second derivative, simply differentiate the function and then differentiate it again

• The function and its first derivative must be differentiable in order to do this

Extending the idea of second derivatives, the function can continue to be differentiated multiple times

• This applies as long as the derivatives continue to be differentiable

The first, second, third and fourth derivatives may be written as

• $f^{\text{'}}\left(x\right),f^{\text{'}\text{'}}\left(x\right),f^{\left(3\right)}\left(x\right),f^{\left(4\right)}\left(x\right)$

• $\frac{dy}{dx},\frac{d^{2}y}{d{x}^2},\frac{d^{3}y}{d{x}^3},\frac{d^{4}y}{d{x}^4}$

• $y^{\text{'}},y^{\text{'}\text{'}},y^{\left(3\right)},y^{\left(4\right)}$

  • You may occasionally see $f^{\text{'}\text{'}\text{'}}\left(x\right)$ or $y^{\text{'}\text{'}\text{'}}$ for the third derivative

  • But beyond the third derivative, the prime notation is almost never used

The third derivative is the:

• rate of change of the rate of change of the rate of change of a function

• This can be extended for higher derivatives if needed

A common use of higher-order derivatives is when describing motion

• If a function describes the displacement of an object,

  • the first derivative describes its velocity

  • the second derivative describes its acceleration

  • the third derivative describes the rate of change of acceleration (sometimes referred to as jerk)

Another common use is when applying L'Hospital's Rule

• $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}=\underset{x\to a}{\lim }\frac{f^{\text{'}\text{'}}\left(x\right)}{g^{\text{'}\text{'}}\left(x\right)}=...$

• As long as the limits continue giving indeterminate forms you can continue applying L’Hospital’s rule with higher order derivatives

• This can make limits far simpler to evaluate