Derivatives Cheat Sheet
Derivative Rules
1. Constant Rule:
2. Power Rule:
d
(c) = 0, where c is a constant
dx
d n
(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
(sin x) = cos x
dx
d
(cos x) = − sin x
dx
d
(tan x) = sec2 x
dx
d
(sec x) = sec x tan x
dx
d
(csc x) = − csc x cot x
dx
d
(cot x) = − csc2 x
dx
Inverse Trigonometric Functions
d
1
(sin−1 x) = √
dx
1 − x2
d
1
(cos−1 x) = − √
dx
1 − x2
d
1
(tan−1 x) =
dx
1 + x2
Exponential & Logarithmic Functions
d x
(a ) = ax ln(a)
dx
d x
(e ) = ex
dx
d
1
(loga (x)) =
dx
x ln(a)
d
1
(ln(x)) =
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.
1.
d n
du
(u ) = nun−1
dx
dx
2.
d u
du
(a ) = au ln(a)
dx
dx
3.
d u
du
(e ) = eu
dx
dx
4.
d
1 du
(loga (u)) =
dx
x ln(u) dx
5.
d
1 du
(ln(u)) =
dx
u dx
6.
d
du
(sin(u)) = cos(u)
dx
dx
7.
d
du
(cos(u)) = − sin(u)
dx
dx
8.
d
du
(tan(u)) = sec2 (u)
dx
dx
9. Same idea for all other trig functions
10.
d
1 du
(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.
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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
, 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)
– Examples:
• 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
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