Why is derivative of constant zero?

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.

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#.

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#