## Derivative Definition:

Measures sensitivity to change
Δ function Δ argument.

## Limit of a Function Notation:

The limit of a function ƒ(x) is L
As L approaches a...

limx → a ƒ(x) = L

## Derivative Notation:

Given a function ƒ(x)
The derivative is ƒ'(x).
Read this as a f prime of x.
Derivative of y in terms of x is:
dy/dx

## Differentiable Definition:

ƒ(x) is differentiable at x = a
if ƒ'(a) exists for each point
in that interval

## Differentiable/Continuous Theorem:

If ƒ(x) is differentiable at
x = a
then ƒ(x) is continous at x = a.

## Derivative of a constant rule:

Derivative of a constant is 0.
ƒ(x) = 6, then ƒ'(x) = 0

## Derivative of a variable:

ƒ(x) = x, then ƒ'(x) = 1

## Derivative power rule:

ƒ(x) = xn, then ƒ'(x) = nxn - 1
ƒ(x) = x2
Using the power rule with n = 2, we have
ƒ'(x) = 2x2 - 1 = 2x

## Derivative Product Rule:

Given a term u = ƒ(x) and
v = g(x)
where
ƒ(x) and g(x) are differentiable
then we have:
d(uv)/dx = u * dv/dx + v * du/dx
This can also be written as
(ƒ(x) * g(x))' = ƒ(x) * g'(x) + g(x) * ƒ'(x)

## Derivative Quotient Rule:

Given a term u = ƒ(x) and
v = g(x)
where
ƒ(x) and g(x) are differentiable
then we have
 (u/v)'  = u' * v + v' * u v2

This can also be written as
 (ƒ(x) / g(x))'  = ƒ'(x) * g(x) + g'(x) * ƒ(x) g(x)2

## Trigonometric Derivatives

ƒ(x)ƒ'(x)Domain
sin(x)cos(x)-∞ < x < ∞
cos(x)-sin(x)-∞ < x < ∞
tan(x)sec2(x)x ≠ π/2 + πn, n ∈ Ζ
csc(x)-csc(x)cot(x)x ≠ πn, n ∈ Ζ
sec(x)sec(x)tan(x)x ≠ π/2 + πn, n ∈ Ζ
cot(x)-csc2(x)x ≠ πn, n ∈ Ζ

ƒ(x)ƒ'(x)
exex
axax * Ln(a)
Ln(x)1/x
logax1/x * Ln(a)

## Derivative Calculator:

For more help, visit out derivative calculator

##### How does the Derivatives Calculator work?
Free Derivatives Calculator - This lesson walks you through the derivative definition, rules, and examples including the power rule, derivative of a constant, chain rule

### What 5 formulas are used for the Derivatives Calculator?

limx → a ƒ(x) = L
ƒ(x) = c, then ƒ'(x) = 0
ƒ(x) = xn then ƒ'(x) = nxn - 1
ƒ(x) * g(x))' = ƒ(x) * g‘(x) + g(x) * ƒ'(x)
(u/v)' = (u' * v + v' * u)/v2

For more math formulas, check out our Formula Dossier

### What 6 concepts are covered in the Derivatives Calculator?

constant
a value that always assumes the same value independent of how its parameters are varied
derivative
rate at which the value y of the function changes with respect to the change of the variable x
function
relation between a set of inputs and permissible outputs
ƒ(x)
limit
the value that a function (or sequence) approaches as the input (or index) approaches some valu
power
how many times to use the number in a multiplication
variable
Alphabetic character representing a number