Derivatives Cheat Sheet - University of Connecticut
Derivatives Cheat Sheet
Derivative Rules
d 1. Constant Rule: (c) = 0, where c is a constant
dx 2. Power Rule: d (xn) = nxn-1
dx 3. Product Rule: (f g) = f g + f g
f f g - fg
4. Quotient Rule:
=
g
g2
5. Chain Rule: (f (g(x)) = f (g(x))g (x)
Common Derivatives
Trigonometric Functions
d (sin x) = cos x
dx d
(sec x) = sec x tan x dx
d (cos x) = - sin x
dx d
(csc x) = - csc x cot x dx
d (tan x) = sec2 x dx d (cot x) = - csc2 x dx
Inverse Trigonometric Functions
d (sin-1 x) = 1
dx
1 - x2
d (cos-1 x) = - 1
dx
1 - x2
d (tan-1 x) dx
=
1
1 + x2
Exponential & Logarithmic Functions
d (ax) = ax ln(a) dx
d (ex) = ex dx
d
1
d
1
dx (loga(x)) = x ln(a)
(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 (un) = nun-1 du
dx
dx
2.
d
(au)
=
au
du ln(a)
dx
dx
3. d (eu) = eu du
dx
dx
d
1 du
4. dx (loga(u)) = 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
8. d (tan(u)) = sec2(u) du
dx
dx
9. Same idea for all other trig functions
10. d (tan-1(u)) = 1 du
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 + y2 = 5, 6xy = 6x + 2y2, 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 (or dy/dx) 3. Solve for y
When finding the second derivative y , remember to replace any y terms in your final answer with the equation for y you already found. In other words, your final answer should not have any y 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
?
Examples:
(x3
+ x) x2 +
cos 1
x ,
ln(x2
+
1)
cos(x)
tan-1(x),
etc.
? 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
3
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