U2: Differentiation: Basic Rules

Identify the form of the derivative first (look at the image above)… is it form a? b? c?
Identify f(x)
Derive f(x) using the corresponding equations next to each form
Plug in the “c” value if applicable.

Example:

lim (2x)3

This is form A.
f(x) = (x)3
Derive → f’(x) = 3x2
Plug in the limit → f’(2) = 3(2)2 = 12

● Differentiable: A function f(x) is differentiable at x = a if f’(a) exists.

Rules of Differentiation

Constant Rule

If the function is just a number, then it would equal 0 because there’s nothing to derive.

Example: f(x) = 5

f’(x) = 0

Constant Multiple Rule

Example: f(x) = 5x4

f’(x) = 5 * 4x3

f’(x) = 20x3

Sum and Difference Rule

Example: f(x) = 3x + 5x3

f’(x) = 3 + 15x2

Power Rule

d/dx [xn] = nx(n - 1)

Example: f(x)=3x2

f’(x) = 2 * 3x(2 - 1)

f’(x) = 6x

Product Rule

f(x) = f(x) * g(x)

f’(x) = f’(x) * g(x) + g’(x) * f(x)

Example: f(x) = 2x2 * 5x

f’(x) = 4x * 5x + 5 * 2x2 = 20x2 + 10x2 = 30x2

Quotient Rule

Example: f(x) = 2/4x

f’(x) = (4x)(0) - (2)(4)/(4x)2 = 0 - 8 / 16x2 = -1/2x2

Derivatives of Trigonometric Functions

● HINT: If the original function starts with C, then the derivative is negative!

○ Example: cosx, cotx, & cscx

Derivative Rule for LN

d/dx ln(x) = 1/x

● HINT: [Derive over copy]

○ Example: h(x) = ln(2x2 + 1)

■ First derive 2x2 + 1. That would be 4x! And then put that over the original function, which would be 2x2 + 1.

■ Your answer would then be 4x/(2x2 + 1)

Deriving Exponential Functions

Continuity

A function f is continuous at “c” if:

● The value exists- The value of the function is defined at “c” and f(c) exists.

● The limit exists - The limit of the function must exist at “c”.

○ The left and right limits must equal.

● Function = limit. The value of the function at “c” must equal the value of the limit at “c”

Discontinuity

● Removable → discontinuity at “c” is called removable if the function can be continuous by defining f(c)

● Non-removable → discontinuity at “c” is called non-removable if the function cannot be made continuous by redefining f(c)

Differentiability

In order for a function to be differentiable at x = c:

● The function must be continuous at x = c

● Its left and right derivative must equal each other at x = c

Example:

f(x) = 2x + 1, x > 2 and x3 - x - 1, x <= 2

● Continuity → 2x + 1 = x3 - x - 1 → 2(2) + 1 = (2)3 - 2 - 1 → 5 = 5

● Differentiability → 2 = 3x2 -1 → 2 = 3(2)2 -1 → 2 ≠11

○ This function is continuous but not differentiable