The derivative of a function f is a new function RULE 2: The POWER Rule
f’.
A function of the form
Derivative is considered to be the slope of a
tangent line, wherein for each x in the domain of f ( x )=x k where k is a real number, is called a power
f, f’(x) is the slope tangent to the graph of f at function.
the point (x, f(x)). Differentiate the given functions.
For y = f(x), the derivative of f at x, denoted by
a) f ( x )=x
f(x), to be
b) f ( x )=x 2
f ( x+ ∆ x )−f ( x)
f ' ( x )= lim
∆x →0 ∆x c) f ( x )=x 3
if the limit exists.
If f ( x )=x n where n ∈ N , then f ' ( x )=n x n−1
Steps in finding the derivative of a function
RULE 3: The CONSTANT MULTIPLE Rule
using the Increment Method:
Find the derivatives of
1. Substitute all values of x’s in the given function
y = f(x) by x + ∆x. a) f ( x )=3 x 2
2. Subtract the given function from the values in
b) f ( x )=5 x 3
step 1 and simplify.
3. Divide the result in step 2 by ∆x. c) f ( x )=4 x 5
4. Take the result in step 3 as ∆x → 0. If f ( x )=k·h(x ) where k is a constant, then
'
f ( x )=k·h ' (x) .
RULE 4: The SUM Rule
The derivative of the function f is the function f’ whose
value at a number x in the domain of f is given by Find the derivative of the following functions.
' f ( x+ h )−f (x) a) f ( x )=4 x +5
f ( x )=lim
h →0 h
b) f ( x )=5 x−3
if the limit exists.
c) f ( x )=x 2 +6 x +2
DIFFERENTIATION RULES
d) f ( x )=2 x 6−3 x 2
f ‘ (x)
dy/dx RULE 5: The PRODUCT Rule
y‘ If f and g are differentiable functions, then
D x [ f (x)· g( x) ] =f (x )g ( x ) + g ( x ) f ' (x ).
'
RULE 1: The CONSTANT Rule
Find the derivative of the following constant functions. Find the derivative of the following functions.
a. f ( x )=10 a) f ( x )=(3 x 2−4 )(x2 −3 x )
b. f ( x )= √ 3 b) f ( x )=(2 x−7)(6 x 3+ 2 x−4)
c. f ( x )=5 π c) f ( x )=(2 x 2+2)(x 2 +3)
If f ( x )=c , where c is a constant, then f ' ( x )=0 . RULE 6: The QUOTIENT Rule
The derivative of a constant is equal to zero. Find the derivative of the following functions.
3 x+ 5
a) f ( x )= 2
x +4
1
b) f ( x )=
2 x+1
2
x +1
c) f ( x )= 2
x −1
Suppose a cat falls from a branch at a given height (ft )
and satisfies the equation h ( t )=16−16 t 2 . What is its
velocity given that it hits the ground at t=1 (s ) ? (32t)
Illustrate and solve the given word problem.
A ball is shot straight up from a building. Its height
(in meters) from the ground at any time t (in seconds) is
given by s ( t )=40+35 t−5 t 2.
Find the instantaneous velocity at t=2.
I have learned that …
I wish to ask my teacher about …
I. Find the derivative of the following functions.
1.) f ( x )=7 x 2 +2
2.) f ( x )=x 4+ 3 x 2−9
II. Solve the given problem.
Calculate the instantaneous velocity of a particle
traveling along a straight line for time t = 3 s with a
function
2
v ( t )=5 t +2 t +3.
Problem: The motion of the car is provided by the
function
2
v ( t )=4 t +10 t +6. Compute its instantaneous velocity
at time t = 5 s.
The Derivative Differentiation
DIFFERENTIATING ALGEBRAIC
FUNCTIONS
Rule 5: The Product Rule
Rule 1: The Derivative of a Constant.
If 𝑓(𝑥) = 𝑐 where c is a constant, then 𝑓′(𝑥) = 0.
The derivative of a constant is equal to zero.
Example 1: If 𝑓(𝑥) = 4
Solution:
𝑓′(𝑥) = 0
Example 2: If ℎ(𝑥) = −√3
Solution:
ℎ′(𝑥) = 0
Example 3: If 𝑔(𝑥) = 5π
Solution:
𝑔′(𝑥) = 0
Rule 2: The Power Rule
The derivative of 𝑥𝑛 is 𝑛𝑥𝑛−1
Example 1: If 𝑓(𝑥) = 𝑥4
Solution:
𝑓′(𝑥) = 4𝑥4−1 = 4𝑥3
Rule 3: The Constant Multiple Rule
If 𝑓(𝑥) = 𝑘𝑓(𝑥) where 𝑘 is a constant, then 𝑓′(𝑥) =
𝑘𝑓′(𝑥).
Example 1: 𝑓(𝑥) = 3𝑥5
Solution:
𝑓′(𝑥) = 3 • (5𝑥5−1) = 15𝑥4
Rule 4: The Sum or a Difference Rule
Rule 6: The Quotient Rule
DIFFERENTIATING EXPONENTIAL AND
LOGARITHMIC FUNCTIONS
Derivatives of Exponential Functions
Exponential functions have the form 𝑓(𝑥) = 𝑎𝑥, where a
is the base. The base is always a positive number not
equal to 1.
Formula:
𝑑/ 𝑑𝑥 (𝑒 𝑢 )=𝑒 𝑢 ∙𝑑𝑢 /𝑑𝑥
Formula: 𝑑 /𝑑𝑥 (𝑎 𝑢 )=𝑎 𝑢 ln𝑎 ∙𝑑𝑢 /𝑑𝑥
Derivatives of Logarithmic Functions
An equation of the form 𝑦 = 𝑙𝑜𝑔𝑎𝑥 where a is any
positive real number except 1 is called logarithmic
function. This function is also defined to be inverse
function of the exponential function.
Formula :
Formula:
DIFFERENTIATING TRIGONOMETRIC AND
INVERSE TRIGONOMETRIC FUNCTIONS
Derivatives of Trigonometric Functions
Formula:
Formula:
Formula s:
Formula s:
