Derivative of Linear Function

In calculus, the derivative is a measure of how a function changes as its input changes. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation.

Linear function refers to a first degree polynomial function. Hence, the derivative of a linear function involves derivative of first degree polynomial function only. A few example problems on derivative of linear function are given below.

(Source: Wikipedia)

Example problems to find derivative of linear function:

Example 1:

Find derivative of a linear function y = 3x + 8

Solution:

Step 1: Given linear function

y = 3x + 8

Step 2: Differentiate the given linear function with respect to ' x '.

(dy)/(dx) = 3

Example 2:

Find derivative of a linear function 4x + 5y = 7

Solution:

Step 1: Given linear function

4x + 5y = 7

Step 2: Subtract 4x on both sides of the equation

4x + 5y - 4x = 7 - 4x

5y = - 4x + 7

Step 3: Divide by 5 on both sides of the equation

Therefore,

y = - 4/5 x + 7/5

Step 4: Differentiate the above linear function with respect to ' x '.

(dy)/(dx)  = - 4/5

Example 3:

Find derivative of a linear function f(x) = x/5 + 9

Solution:

Step 1: Given linear function

f(x) = x/5 + 9

Step 2: Differentiate the given linear function with respect to ' x '.

f'(x) = 1/5

Example 4:

Find second order derivative of a linear function y = 7x + 10

Solution:

Step 1: Given linear function

y = 7x + 10

Step 2: Differentiate the given linear function with respect to ' x '.

(dy)/(dx) = 7

Step 3: Again differentiate the above function with respect to ' x '.
(d^2y)/(dx^2) = 0



Example 5:

Find (dy)/(dx)  of the function - 5x + 5y = 7

Solution:

Step 1: Given linear function

- 5x + 5y = 7

Step 2: Add 5x on both sides of the given linear equation

- 5x + 5y + 5x = 7 + 5x

5y = 5x + 7

Step 3: Divide by 5 on both sides

We get,

y = x + 7/5

Step 4: Differentiate the above linear function with respect to ' x ' to find (dy)/(dx)

(dy)/(dx) = 1

Practice problems to find derivative of a linear function:

1) Find derivative of a linear function y = 5x + 13

2) Find derivative of a linear function 3x - 7y = 6

3) Find second order derivative of a linear function y = 8x + 15

4) Find derivative of a linear function f(x) = - 6/7 x + 9

5) Find (dy)/(dx) for y = 11x + 8

Solutions:

1) (dy)/(dx) = 5

2) (dy)/(dx) = 3/7

3) (d^2y)/(dx^2) = 0

4) f'(x) = - 6/7

5) (dy)/(dx) = 11