| Derivative of a Function |

Nov. 28, 2024, 6:55 p.m.

Definition

 

Derivative of a Function

-> the limit of the ratio of the increment function to the increment of independent variable when it varies and approaches zero.

-> also known as instantaneous rate of change.

Symbol:

Where:

The derivative of a function f at a number x.

 

General Rules for Differentiation

 

1. In the original equation, replace x with , and y with , and simplify.

2. Subtract the original equation from the new equation, thus obtaining in terms of x and and simplify.

3. Divide both members of the equation by and simplify.

4. Find the limit of both sides of the equation as .

 

Example

 

1. Find the answer from this given:

 

Steps:

a. Substitute the x with and .

b. Rearrange with the same value, where the must be at the left, while the rest at the right side.

c. Substitute the y equivalent.

d. Factor out the .

e. Substitute the with 0.

f. Find the limit.

=  

=

=

=

=

=

= 3x2 - 2

 

Exercises

 

1. x = y4 - 2y3

Steps:

a. Use the derivative of a function. 

b. Substitute the x and y with and .

c. Rearrange with the same value, where the must be at the left, while the rest at the right side.

d. Substitute the x equivalent.

e. Factor out the .

f. Substitute the with 0.

g. Find the limit.

 

Solution:

= x = y4 - 2y3

=

=

=

=

=

=

= 4y3 - 6y2

 

Answer: 4y3 - 6y2

2.

Steps:

a. Use the derivative of a function. 

b. Substitute the x and y with and .

c. Rearrange with the same value, where the must be at the left, while the rest at the right side.

d. Substitute the y equivalent.

e. Factor out the .

f. Substitute the with 0.

g. Find the limit.

 

Solution:

=

=

=

=

=

=

=

 

Answer:

3.

Steps:

a. Use the derivative of a function. 

b. Substitute the m and n with and m with .

c. Rearrange with the same value, where the must be at the left, while the rest at the right side.

d. Substitute the m equivalent.

e. Factor out the .

f. Substitute the with 0.

g. Find the limit.

 

Solution:

=

=

=  

=

=

=

=

=

 

Answer:

4.

Steps:

a. Use the derivative of a function. 

b. Substitute the a and b with and .

c. Rearrange with the same value, where the must be at the left, while the rest at the right side.

d. Substitute the b equivalent.

e. Factor out the .

f. Substitute the with 0.

g. Find the limit.

 

Solution:

=

=

=

=

=

=

=

=

=

 

Answer:

5. g = cos h

Steps:

a. Use the derivative of a function. 

b. Substitute the g and h with and .

c. Rearrange with the same value, where the must be at the left, while the rest at the right side.

d. Substitute the g equivalent.

e. Factor out the .

f. Substitute the with 0.

g. Find the limit.

 

Solution:

= g = cos h

=

=  

=

=

=

=

=

=

=

= - sin h

 

Answer: - sin h