Welcome to a
Proof of The Constant Derivative Rule,
which is the derivative of a constant, c,
with respect to x equals zero.
You'll be using the
Definition of the Derivative
given here below, where
f prime of x equals
the limit as h approaches zero
of the difference quotient.
Some textbooks use delta x instead of h,
we'll be using the
definition in this form.
So again, our goal here is
to prove the derivative of
a constant with respect
to x is equal to zero.
We know the derivative is
equal to this limit definition.
In this case, the function f of x
is equal to the constant c,
and therefore for any input,
the output is going to be the constant c.
So, f of the quantity
x plus h, and f of x,
would both be equal to c,
and therefore, our limit
simplifies to the limit
as h approaches zero
of the quantity c minus c divided by h.
And of course, c minus c is equal to zero.
And zero divided by h,
as long as h isn't zero,
simplifies to just zero, and therefore,
the limit as h approaches
zero of zero, is just zero
proving the derivative of a constant
with respect to x is zero.
