C.2 Derivatives

A complete review on derivatives would be lengthy. We try to touch on some topics that are used often in STAT 414 but not everything can be covered in the review. There are many good calculus books and websites to help you review. Students like the book Forgotten Calculus: A Refresher Course with Applications to Economics and Business by Barbara Lee Bleau, Ph.D. as a reference.

The definition of a derivative is

\[f^\prime(x)=\frac{d}{dx} f(x)=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}\]

The derivative is the slope of the tangent line to the graph of \(f(x)\), assuming the tangent line exists. You can find further explanations of derivatives on the web using websites like Khan Academy. Below are rules for determining derivatives and links for extra help.

  1. Common Derivatives and Rules
    1. Power Rule:
      \(\frac{d}{dx}x^n=nx^{n-1}\) (Power Rule, Khan Academy)
    2. \(\frac{d}{dx} \ln x=\frac{1}{x}\)
    3. \(\frac{d}{dx} a^x=a^x\ln a\)
    4. \(\frac{d}{dx} e^x=e^x\)
  2. Product rule 
    \(\begin{equation}\left[f(x)g(x)\right]^\prime=f^\prime(x)g(x)+f(x)g^\prime(x)\end{equation}\) (Product Rule, Khan Academy)
  3. Quotient rule
    \(\begin{equation} \left[\frac{f(x)}{g(x)}\right]^\prime=\frac{g(x)f^\prime(x)-f(x)g^\prime(x)}{\left(g(x)\right)^2}\end{equation}\) (Quotient Rule, Khan Academy)
  4. Chain Rule
    Let \(y=f(g(x))\) where f and g are functions, g is differentiable at x, and f is differentiable at \(g(x)\). Then the derivative of y is \(f^\prime(g(x))g^\prime(x)\). (Chain Rule, Khan Academy)
  5. L'Hopital's Rule
    1. For the type \(0/0\): Suppose that \(\lim_{x\rightarrow u} f(x)=0\) and \(\lim_{x\rightarrow u} g(x)=0\). If \(\lim_{x\rightarrow u}\left[\frac{f^\prime(x)}{g^\prime(x)}\right]\) exists in either the finite or infinite sense, then\[\begin{equation}\lim_{x\rightarrow u} \frac{f(x)}{g(x)}=\lim_{x\rightarrow u} \frac{f^\prime(x)}{g^\prime(x)}=\frac{\lim_{x\rightarrow u} f^\prime(x)}{\lim_{x\rightarrow u} g^\prime(x)}\end{equation}\]
    2. For the type \(\infty/\infty\): Suppose that \(\lim_{x\rightarrow u} |f(x)|=\infty\) and \(\lim_{x\rightarrow u} |g(x)|=\infty\). If \(\lim_{x\rightarrow u}\left[\frac{f^\prime(x)}{g^\prime(x)}\right]\) exists in either the finite or infinite sense, then\[\begin{equation}\lim_{x\rightarrow u} \frac{f(x)}{g(x)}=\lim_{x\rightarrow u} \frac{f^\prime(x)}{g^\prime(x)}=\frac{\lim_{x\rightarrow u} f^\prime(x)}{\lim_{x\rightarrow u} g^\prime(x)}\end{equation}\]
    3. Other indeterminate forms can also be solved using L'Hopital's Rule, such as \(0^0\) and \(\infty^0\). It would be a good idea for review the uses of L'Hopital's Rule. (L'Hopital's Rule, Khan Academy)

Example C.2.1

Find the derivative of \(f(x)\) for the following:

  1. \(f(x)=10x^9-5x^5+7x^3-9\)
  2. \(f(x)=\dfrac{x}{x^2+5}\)
  3. \(f(x)=\dfrac{1}{\sqrt{x}}\)
  1. \(f^\prime(x)=90x^8-25x^4+21x^2\)
  2. Using the Quotient Rule, \(f^\prime(x)=\dfrac{x^2+5-x(2x)}{(x^2+5)^2}=\dfrac{-x^2+5}{(x^2+5)^2}\)
  3. Using the Power Rule, \(f^\prime(x)=-\dfrac{1}{2x\sqrt{x}}\)

Example C.2.2

Find the derivative of \(f(x)=-\ln x^2\).
Using the Chain Rule, \(f^\prime(x)=-\dfrac{2}{x}\)