Q5 - Calculus

Question

Why does it make graphical sense that the derivative of a constant is zero? That the derivative of the identity function is constantly equal to 1? That the derivative of a linear function f (x) = mx + b is equal to m  ?

Step-by-Step Solution

Answer

The graph of derivative of constant function is a horizontal line that is y axis .

Derivative of identity function is 1. The rate of change of y with respect to x is 1.

Derivative of linear function is 'm'. The rate of change of y with respect to x is 'm'

Step 1: Given Information

the derivative of a constant is zero 

 the derivative of the identity function is constantly equal to 1 

the derivative of a linear function f (x) = mx + b is equal to m 

Step 2: Constant function

Consider a function f(x) = k

Derivative of a constant function is always 0 

f'(x)=0

y=0

The graph of y=0 is a horizontal line that is y axis .

Step 3: Identity function

Lets consider and identity function 

f(x)=x

Derivative of f(x)=x  is 1

Derivative of x is 1. 

Derivative of identity function is 1. The rate of change of y with respect to x is 1.

Step 4: Linear function

consider a linear function f(x)=mx+b

When we differentiate with respect to x

f'(x)=m

Derivative of linear function is 'm'. The rate of change of y with respect to x is 'm'