Derivative Toolbox Derivative Toolbox
Introduction
For the next few lessons, we will be building our derivative toolbox. The first tools we
will learn are the following:
f(x)=a \quad f'(x)=0
The derivative of a constant is zero.
f(x)=a\,g(x) \quad f'(x)=a\,g'(x)
The derivative of a constant and function is equal to the constant and the derivative of the function.
f(x)=g(x) + j(x) \quad f'(x)=g'(x) + j'(x)
The derivative of a sum of functions is the sum of the derivatives of the functions.
Proving the Derivative of a Constant
f(x)=a
Define the derivative.
f'(x)=\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}
Substitute
f(x)
with
a
.
f'(x)=\lim_{h \to 0}\frac{a - a}{h}
Remove
a
and
-a
f'(x)=\lim_{h \to 0}\frac{0}{h}
Solve the limit.
f'(x)=0
Proving the Derivative of a Constant Times a Function
f(x)=a\,g(x)
Define the derivative. g(x) represents any function.
f'(x)=\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}
Substitute
f(x)
with
a\,g(x)
.
f'(x)=\lim_{h \to 0}\frac{a\,g(x+h) - a\,g(x)}{h}
Extract
a
from the function.
f'(x)=\lim_{h \to 0}\frac{a(g(x+h) - g(x))}{h}
Extract
a
from the limit.
f'(x)=a\,\lim_{h \to 0}\frac{g(x+h) - g(x)}{h}
Substitute the derivative of
g(x)
with
g'(x)
.
f'(x)=a\,g'(x)
Proving the Derivative of a Sum of Functions
f(x)=g(x) + j(x)
Define the derivative. (g(x) represents any function.)
f'(x)=\lim_{h \to 0}\frac{f(x+h) - f(x)}{h}
Substitute
f(x)
with
g(x) + j(x)
.
f'(x)=\lim_{h \to 0}\frac{g(x+h) + j(x+h) - (g(x) + j(x))}{h}
Rearrange the terms.
f'(x)=\lim_{h \to 0}\frac{(g(x+h) - g(x)) + (j(x+h) - j(x))}{h}
Separate the fraction.
f'(x)=\lim_{h \to 0}\frac{g(x+h) - g(x)}{h} + \frac{j(x+h) - j(x)}{h}
Separate the limits.
f'(x)=\lim_{h \to 0}\frac{g(x+h) - g(x)}{h} + \lim_{h \to 0}\frac{j(x+h) - j(x)}{h}
Substitute the derivatives of
g(x)
and
j(x)
with
g'(x)
and
j'(x)
.
f'(x)=g'(x) + j'(x)