Higher Order Derivatives (DP IB Analysis & Approaches (AA): HL): Revision Note

Second order derivatives

What is a second order derivative?

• If you differentiate the derivative of a function (i.e. differentiate the function a second time) you get the second order derivative of the function

• There are two forms of notation for the second order derivative

  • $y=f\left(x\right)$

  • $\frac{\mathrm{d}y}{\mathrm{d}x}=f^{\text{'}}\left(x\right)$     (First order derivative) 

  • $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=f^{\text{'}\text{'}}\left(x\right)$     (Second order derivative)

• Note the position of the superscript 2’s

  • differentiating twice (so${\mathbf{d}}^{\mathbf{2}}$) with respect to $x$ twice (so${\mathit{x}}^{\mathbf{2}}$)

• The second order derivative can be referred to simply as the second derivative

  • Similarly, the first order derivative can be called just the first derivative

• A first order derivative is the rate of change of a function

  • a second order derivative is the rate of change of the rate of change of a function

    • i.e. the rate of change of the function’s gradient

• Second order derivatives can be used to

  • test for local minimum and maximum points

  • help determine the nature of stationary points

  • determine the concavity of a function

  • help graph the derivative of a function

How do I find a second order derivative of a function?

• By differentiating twice!

• This may involve

  • rewriting fractions, roots, etc. as negative and/or fractional powers

  • differentiating trigonometric functions, exponentials and logarithms

  • using the chain rule

  • using the product or quotient rules

Higher order derivatives

What is meant by higher order derivatives of a function?

• Many functions can be differentiated numerous times

  • The third, fourth, fifth, etc derivatives of a function are generally called higher order derivatives

• It may not be possible, or practical to (algebraically) differentiate complicated functions more than once or twice

• Polynomials will, eventually, have higher order derivatives of zero

  • Since powers of x reduce by 1 each time

What is the notation for higher order derivatives?

• The notation for higher order derivatives follows the logic from the first and second derivatives

  • $f^{\left(n\right)}\left(x\right)$  or $\frac{{\mathrm{d}}^{n}y}{\mathrm{d}{x}^n}$for the $n$th derivative

    • So the fifth derivative would be $f^{\left(5\right)}\left(x\right)$ or $\frac{{\mathrm{d}}^{5}y}{\mathrm{d}{x}^5}$

• The ‘dash’ notation is replaced with numbers after a point, as it would become cumbersome after the first few

  • So you could use $f^{\text{'}}\left(x\right)$, $f^{\text{'}\text{'}}\left(x\right)$ and $f^{\text{'}\text{'}\text{'}}\left(x\right)$ for the first three derivatives

  • but after that $f^{\left(4\right)}\left(x\right)$, $f^{\left(5\right)}\left(x\right)$, etc. should be used for the 4th, 5th, etc., derivatives

    • $f^{\left(3\right)}\left(x\right)$ for the third derivative is also common

How do I find a higher order derivative of a function?

• By differentiating as many times as required!

• This may involve

  • rewriting fractions, roots, etc. as negative and/or fractional powers

  • differentiating trigonometric functions, exponentials and logarithms

  • using the chain rule

  • using the product or quotient rules