Why is derivative of constant zero?

3 Answers
Dec 22, 2015

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

Take and graph the constant 44:
graph{0x+4 [-9.67, 10.33, -2.4, 7.6]}

The constant never changes—it is constant.

Thus, the derivative will always be 00.

Consider the function x^2-3x23.
graph{x^2-3 [-9.46, 10.54, -5.12, 4.88]}

It is the same as the function x^2x2 except that it's been shifted down 33 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 2x2x. When finding the derivative of x^2-3x23, the -33 can be disregarded since it does not change the way in which the function changes.

Dec 22, 2015

Use the power rule: d/dx[x^n]=nx^(n-1)ddx[xn]=nxn1

A constant, say 44, can be written as

4x^04x0

Thus, according to the power rule, the derivative of 4x^04x0 is

0*4x^-104x1

which equals

00

Since any constant can be written in terms of x^0x0, finding its derivative will always involve multiplication by 00, resulting in a derivative of 00.

Dec 22, 2015

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