Scribd
Upload
8K views3 pages

Derivatives Cheat Sheet

This document provides a cheat sheet on derivatives, including the basic rules of differentiation as well as common derivatives of trigonometric, inverse trigonometric, exponential, and logarithmic functions. It also covers techniques like the chain rule, implicit differentiation, and log differentiation that are useful for taking derivatives of more complex expressions.

Uploaded by

alex
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
8K views3 pages

Derivatives Cheat Sheet

This document provides a cheat sheet on derivatives, including the basic rules of differentiation as well as common derivatives of trigonometric, inverse trigonometric, exponential, and logarithmic functions. It also covers techniques like the chain rule, implicit differentiation, and log differentiation that are useful for taking derivatives of more complex expressions.

Uploaded by

alex
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 3

Derivatives Cheat Sheet

Derivative Rules
d
1. Constant Rule: (c) = 0, where c is a constant
dx
d n
2. Power Rule: (x ) = nxn−1
dx
3. Product Rule: (f g)0 = f 0 g + f g 0
 0
f f 0g − f g0
4. Quotient Rule: =
g g2
5. Chain Rule: (f (g(x))0 = f 0 (g(x))g 0 (x)

Common Derivatives

Trigonometric Functions

d d d
(sin x) = cos x (cos x) = − sin x (tan x) = sec2 x
dx dx dx
d d d
(sec x) = sec x tan x (csc x) = − csc x cot x (cot x) = − csc2 x
dx dx dx

Inverse Trigonometric Functions

d 1 d 1 d 1
(sin−1 x) = √ (cos−1 x) = − √ (tan−1 x) =
dx 1 − x2 dx 1 − x2 dx 1 + x2

Exponential & Logarithmic Functions

d x d x
(a ) = ax ln(a) (e ) = ex
dx dx
d 1 d 1
(loga (x)) = (ln(x)) =
dx x ln(a) dx x

1
Chain Rule
In the below, u = f (x) is a function of x. These rules are all generalizations of the above rules using the
chain rule.
d n du
1. (u ) = nun−1
dx dx
d u du
2. (a ) = au ln(a)
dx dx
d u du
3. (e ) = eu
dx dx
d 1 du
4. (loga (u)) =
dx x ln(u) dx
d 1 du
5. (ln(u)) =
dx u dx
d du
6. (sin(u)) = cos(u)
dx dx
d du
7. (cos(u)) = − sin(u)
dx dx
d du
8. (tan(u)) = sec2 (u)
dx dx
9. Same idea for all other trig functions
d 1 du
10. (tan−1 (u)) =
dx 1 + u2 dx
11. Same idea for all other inverse trig functions

Implicit Differentiation
Use whenever you need to take the derivative of a function that is implicitly defined (not solved for y).
Examples of implicit functions: ln(y) = x2 , x3 + y 2 = 5, 6xy = 6x + 2y 2 , etc.

Implicit Differentiation Steps:

1. Differentiate both sides of the equation with respect to “x”

2. When taking the derivative of any term that has a “y” in it multiply the term by y 0 (or dy/dx)

3. Solve for y 0

When finding the second derivative y 00 , remember to replace any y 0 terms in your final answer with the
equation for y 0 you already found. In other words, your final answer should not have any y 0 terms in it.

2
Log Differentiation
Two cases when this method is used:

• Use whenever you can take advantage of log laws to make a hard problem easier

(x3 + x) cos x
– Examples: , ln(x2 + 1) cos(x) tan−1 (x), etc.
x2 + 1
– Note that in the above examples, log differentiation is not required but makes taking the
derivative easier (allows you to avoid using multiple product and quotient rules)

• Use whenever you are trying to differentiate


d
f (x)g(x)

dx

√
– Examples: xx , x x
, (x2 + 1)x , etc.
– Note that in the above examples, log differentiation is required. There is no other way to take
these derivatives.

Log Differentiation Steps:

1. Take the ln of both sides

2. Simplify the problem using log laws

3. Take the derivative of both sides (implicit differentiation)

4. Solve for y 0

You might also like