Derivatives

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...

lim_{x → 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) = nx^{n - 1}

ƒ(x) = x²
Using the power rule with n = 2, we have
ƒ'(x) = 2x^{2 - 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

Derivative Calculator:

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