Derivatives

What are Derivatives?

A quick introduction to what derivatives are so that everything else makes sense.

• Quick Refresher
  Rate of Change

  Derivatives are a method for analyzing the instantaneous rate of change of a function.
• Lesson
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Calculus: Building Intuition for the Derivative	BetterExplained	Article	Medium
  Derivative Notation Review	Khan Academy	Article	Short
  What is a Derivative? Understanding the Definition	PatrickJMT	Video	12:08

What is the general logic that allows a rather simple program to compute infinitely many derivatives?

• Quick Refresher
  Simplify and Repeat

  The computer knows how to differentiate a few basic functions. If it runs into a function that it doesn't know how to differentiate then it applies a rule to break up the function into simpler parts. This process is repeated until every part is fully differentiated.
• Lesson
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Tips for Derivatives	Brown University	Article	Short
  Derivatives in 20 minutes	Carter Bowles	Video	20:29
  How to Take Derivatives	WikiHow	Article	Medium

The Power Rule

The backbone of differentiation is the remarkably simple power rule. Understanding it fully is key.

• Quick Refresher
  Apply when x is raised to a constant power.

  Bring the exponent to the front and reduce the exponent by one.
  $$\frac{d}{dx}[x^n]=nx^{n-1}$$
• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  The Power Rule with animations...	Math Scoop	Interactive	Short
  Derivatives – Basic Examples	PatrickJMT	Video	9:07
  Proof of the Power Rule...	Khan Academy	Video	7:02

Exponentials and Logarithms

Two more functions that appear repeatedly in any Calculus course and have easy derivatives.

• Quick Refresher
  The basic forms

  $$\frac{d}{dx}[e^x]=e^x$$ $$\frac{d}{dx}[\ln{(x)}]=\frac{1}{x}$$
• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Derivatives of Exponential and Logarithm Functions	Paul's Notes	Article	Medium
  Derivatives of Exponential & Logarithmic Functions	Brown University	Article	Short
  Derivatives of Exponential Functions	PatrickJMT	Video	4:35

Trigonometric Functions

Learn how to differentiate this versatile family of functions. You want to memorize these derivatives.

• Quick Refresher
  Learn all 12 derivatives

  $f(x)$	$\frac{d}{dx}[f(x)]$
  $\sin{(x)}$	$\cos{(x)}$
  $\cos{(x)}$	$-\sin{(x)}$
  $\tan{(x)}$	$\sec{(x)}^2$
  $\cot{(x)}$	$-\csc{(x)}^2$
  $\sec{(x)}$	$\sec{(x)}\tan{(x)}$
  $\csc{(x)}$	$-\csc{(x)}\cot{(x)}$

  $f(x)$	$\frac{d}{dx}[f(x)]$
  $\sin^{-1}{(x)}$	$\frac{1}{\sqrt{-x^2+1}}$
  $\cos^{-1}{(x)}$	$\frac{-1}{\sqrt{-x^2+1}}$
  $\tan^{-1}{(x)}$	$\frac{1}{x^2+1}$
  $\cot^{-1}{(x)}$	$\frac{-1}{x^2+1}$
  $\sec^{-1}{(x)}$	$\frac{1}{|x|\sqrt{x^2-1}}$
  $\csc^{-1}{(x)}$	$\frac{-1}{|x|\sqrt{x^2-1}}$

• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Derivatives of Trigonometric Functions	Brown University	Article	Short
  How to Remember Derivatives of Trig...	Verdant Media	Video	6:43
  Differentiation of trigonometric functions	Wikipedia	Article	Long

Sum Rule and Constant Multiples

The easiest rule in Calculus is the sum rule so make sure you understand it.

• Quick Refresher
  Differentiate each term

  Separate the function into its terms and find the derivative of each term. Then add up the derivatives. Also applies to subtraction in the same way. Constant multiples are a specific case of the sum rule. $$\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}[f(x)]+\frac{d}{dx}[g(x)]$$ $$\frac{d}{dx}[f(x)-g(x)]=\frac{d}{dx}[f(x)]-\frac{d}{dx}[g(x)]$$ $$\frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]$$
• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Differentiating Polynomials	Khan Academy	Video	9:52
  Sum Rule Proof	Math2	Article	Short
  Sum Rule	Wikipedia	Article	Medium

Product Rule

Products are not as nice as sums, but learn the rule and differentiate any product.

• Quick Refresher
  Determine the factors

  Rewrite the function as the product of two simpler functions. Then differentiate these two functions and combine them as dictated by the product rule. $$\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}[g(x)]+g(x)\frac{d}{dx}[f(x)]$$
• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Product and Quotient Rule	Paul's Online Math Notes	Article	Medium
  Product Rule for Derivatives	Harvey Mudd College	Article	Short
  Product Rule Intro	Khan Academy	Video	2:39

Quotient Rule

The quotient rule is as straight-forward as the product rule, but you need to memorize a little bit more.

• Quick Refresher
  Differentiate Fractions

  Rewrite the function as a fraction with numerator and denominator each dependent on $x$. Differentiate the numerator and denominator and input these derivatives into thte quotient rule. $$\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)\frac{d}{dx}[f(x)]-f(x)\frac{d}{dx}[g(x)]}{g(x)^2}$$
• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Quotient Rule Made Easier	mathbff	Video	3:13
  Quotient Rule for Derivatives	Harvey Mudd College	Article	Short
  The Quotient Rule	PatrickJMT	Video	7:37

Chain Rule

If nothing else works then you're likely dealing with a composition of functions requiring the chain rule.

• Quick Refresher
  Undo Function Composition

  First, determine the outside function. Then what is left over is the inside function. Determine the derivative of the outside and inside functions and input into the chain rule. $$\frac{d}{dx}[f(g(x))]=\frac{d}{dg(x)}[f(g(x))]\frac{d}{dx}[g(x)]$$
• Lesson
• Worksheet
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Chain Rule	Paul's Online Math Notes	Article	Medium
  Chain Rule for Finding Derivatives	PatrickJMT	Video	4:14
  The Chain Rule...How?	mathbff	Video	16:36

Logarithmic Differentiation

One final type of function requires logarithmic differentiation.

• Quick Refresher
  Take a logarithm before differentiating

  Logarithmic differentiation is the process of differentiating the natural logarithm of a function rather than the function itself. Obviously this method only helps in specific cases, but the derivative can be obtained with the following rule: $$\frac{d}{dx}[f(x)]=\frac{d}{dx}[\ln(f(x))]f(x)$$.
• Lesson
• External Links
  Quality links from around the web.

  Title	Source	Format	Length
  Logarithmic Differentiation	Paul's Online Math Notes	Article	Medium
  Logarithmic Differentiation	PatrickJMT	Video	8:28
  Logarithmic Differentiation	UC Davis	Article	Short