Derivatives Worksheets - Learn to Differentiate with Calculus College

One

The Power Rule

It shouldn't take you long to work power rule problems of all types.

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

Two

Exponentials and Logarithms

There are only a few functions to deal with so get some practice with all of them.

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

Three

Trigonometric Functions

Test whether you have actually memorized all of these derivatives so you're not overconfident.

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

Four

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

Five

Product Rule

Products are not as nice as sums, so spend some time practicing how to keep track of all the functions.

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

Six

Quotient Rule

The more quotient rule problems you work the more natural it will be and you won't have to second guess your memory.

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

Seven

Chain Rule

The chain rule needs to be explored with practice so get to work on it right now.

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

Derivatives

More Derivatives

Derivatives

Derivative Worksheets

One

The Power Rule

Two

Exponentials and Logarithms

Three

Trigonometric Functions

Four

Sum Rule and Constant Multiples

Five

Product Rule

Six

Quotient Rule

Seven

Chain Rule