Welcome to a
Proof of the Constant
Multiple Derivative Rule,
which is the derivative
of a constant times the
function f of x with respect to x equals
the constant c times f prime of x.
We'll be using the Definition
of the Derivative shown here,
where f prime of x equals the limit
as h approaches zero of
the difference quotient.
Notice how, for this
definition, we're using h.
Some textbooks will use delta x instead.
To begin our proof, if we apply
the limit definition of the
derivative to c times f of x,
we would have the limit
as h approaches zero of c
times f of the quantity x plus h minus c
times f of x all divided by h.
Notice now we have a common
factor of c in the numerator.
So, the next step is to factor.
So, notice now here we factored out the c
from the numerator, and now we can
factor this constant c out of the limit.
So, now we have c times the limit
as h approaches zero of, this is just the
difference quotient, and
therefore, this limit is equal to
f prime of x, giving us
c times f prime of x.
Now we have the proof
that the derivative of
a constant c times f
of x with respect to x
equals c times f prime of x.
