Question

Why is derivative of constant zero?

Answer 1

The derivative represents the change of a function at any given time.

Take and graph the constant `4`:
graph{0x+4 [-9.67, 10.33, -2.4, 7.6]}

The constant never changes—it is constant.

Thus, the derivative will always be `0`.

Consider the function `x^2-3`.
graph{x^2-3 [-9.46, 10.54, -5.12, 4.88]}

It is the same as the function `x^2` except that it's been shifted down `3` units.
graph{x^2 [-9.46, 10.54, -5.12, 4.88]}

The functions increase at exactly the same rate, just in a slightly different location.

Thus, their derivatives are the same—both `2x`. When finding the derivative of `x^2-3`, the `-3` can be disregarded since it does not change the way in which the function changes.

Answer 2

Use the power rule: `d/dx[x^n]=nx^(n-1)`

A constant, say `4`, can be written as

`4x^0`

Thus, according to the power rule, the derivative of `4x^0` is

`0*4x^-1`

which equals

`0`

Since any constant can be written in terms of `x^0`, finding its derivative will always involve multiplication by `0`, resulting in a derivative of `0`.

Answer 3

Use the limit definition of the derivative:

`f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h`

If `f(x)="C"`, where `"C"` is any constant, then

`f(x+h)="C"`

Thus,

`f'(x)=lim_(hrarr0)("C"-"C")/h=lim_(hrarr0)0/h=lim_(hrarr0)0=0`