C.2 Derivatives - STAT ONLINE

Skip to main content

Breadcrumb

1. Home
2. reviews
3. calculus
4. derivatives

C.2 Derivatives

A complete review of 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. \(f(x)=\dfrac{1}{\sqrt{x}}\)

Answer

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

• « PreviousC.1 Summations and Series
• NextC.3 Integrals »

Statistics Online

• For Students
  • Learning Online Orientation
    • O.1 How do the online courses work?
    • O.2 What technology is used?
      • Technical Requirements for Online Courses
      • Learning Management System: Canvas
      • Take a Screen Capture
    • O.3 What is a proctored exam?
    • O.4 How can I be successful?
    • O.5 What resources are available?
    • O.6 Summary
  • Math & Stat Reviews
    • Algebra Review
      • A.1 Order of Operations
      • A.2 Summations
      • A.3 Factorials
      • A.4 Self-Assess
    • Basic Statistical Concepts
      • S.1 Basic Terminology
      • S.2 Confidence Intervals
      • S.3 Hypothesis Testing
        • S.3.1 Hypothesis Testing (Critical Value Approach)
        • S.3.2 Hypothesis Testing (P-Value Approach)
        • S.3.3 Hypothesis Testing Examples
      • S.4 Chi-Square Tests
      • S.5 Power Analysis
      • S.6 Test of Proportion
      • S.7 Self-Assess
    • Calculus Review
      • C.1 Summations and Series
      • C.2 Derivatives
      • C.3 Integrals
      • C.4 Multivariable Calculus
    • Matrix Algebra Review
      • M.1 Matrix Definitions
      • M.2 Matrix Arithmetic
      • M.3 Matrix Properties
      • M.4 Matrix Inverse
      • M.5 Advanced Matrix Properties
      • M.6 Range, Nullspace and Projections
      • M.7 Gauss-Jordan Elimination
      • M.8 Eigendecomposition
      • M.9 Self-Assess
  • Ethics and Statistics
    • Scholarship and Research Integrity
    • The Penn State Values
    • ASA Ethics Guidelines
    • P-STAT Accreditation
    • Ethics Scenarios

• Technology Tutorials
  • Communicating Online
    • Discussion
    • Video Conferencing
  • Statistical Software
    • Minitab
    • PSU WebLabs
    • R
    • SAS
    • SPSS
  • Working with Digital Content
    • Cloud Storage
    • Creating Videos
    • Math Type Online
    • Scanning and Uploading

× Save changes Close