<handout>
<title>Guided Notes: Derivatives of sums</title>
<page>
<p workspace="1in">
Today we will explore how to take the derivative of the sum of two functions.
For example, if
<me>
f(x) = x^2 + 3x
</me>,
what is <m>f'(x)</m>?
What two functions is this the sum of?
What are the derivatives of each of those functions?
</p>
<p>
To be sure of the derivative of the sum,
we should use the definition of the derivative.
</p>
<definition workspace=".75in">
<title>Definition of the Derivative</title>
<statement>
<p>
The derivative of a function <m>f(x)</m> at any point <m>x</m> is defined as...
</p>
</statement>
</definition>
<p workspace="2in">
Now let's apply this definition to the function <m>f(x) = x^2 + 3x</m>.
We have:
</p>
<p workspace="0.25in">
What should the general rule be?
</p>
</page>
<page>
<theorem workspace="0.5in">
<statement>
<p>
For any two differentiable functions <m>f(x)</m> and <m>g(x)</m>,
the derivative of their sum <m>h(x) = f(x) + g(x)</m> is given by:
</p>
</statement>
<proof workspace="3in">
<p>
Let <m>f(x)</m> and <m>g(x)</m> be differentiable functions and let <m>h(x) = f(x) + g(x)</m>.
Then by the limit definition of the derivative,
</p>
</proof>
</theorem>
<example workspace="1in">
<statement>
<p>
Find the derivative of <m>f(x) = x^5 + e^x</m>.
</p>
</statement>
</example>
<example workspace="1in">
<statement>
<p>
Find the derivative of <m>f(x) = \sqrt{x} + x^3 + 7x</m>.
</p>
</statement>
</example>
<example workspace="1in">
<statement>
<p>
Find the derivative of <m>f(x) = 5x^4</m>.
</p>
</statement>
</example>
</page>
</handout>
Print preview
Handout 36.2 Guided Notes: Derivatives of sums
View Source for handout
Today we will explore how to take the derivative of the sum of two functions. For example, if
\begin{equation*}
f(x) = x^2 + 3x\text{,}
\end{equation*}
what is \(f'(x)\text{?}\) What two functions is this the sum of? What are the derivatives of each of those functions?
To be sure of the derivative of the sum, we should use the definition of the derivative.
Definition 36.1. Definition of the Derivative.
View Source for definition
<definition workspace=".75in">
<title>Definition of the Derivative</title>
<statement>
<p>
The derivative of a function <m>f(x)</m> at any point <m>x</m> is defined as...
</p>
</statement>
</definition>
Now let’s apply this definition to the function \(f(x) = x^2 + 3x\text{.}\) We have:
What should the general rule be?
Theorem 36.2.
View Source for theorem
<theorem workspace="0.5in">
<statement>
<p>
For any two differentiable functions <m>f(x)</m> and <m>g(x)</m>,
the derivative of their sum <m>h(x) = f(x) + g(x)</m> is given by:
</p>
</statement>
<proof workspace="3in">
<p>
Let <m>f(x)</m> and <m>g(x)</m> be differentiable functions and let <m>h(x) = f(x) + g(x)</m>.
Then by the limit definition of the derivative,
</p>
</proof>
</theorem>
For any two differentiable functions \(f(x)\) and \(g(x)\text{,}\) the derivative of their sum \(h(x) = f(x) + g(x)\) is given by:
Proof.
View Source for proof
<proof workspace="3in">
<p>
Let <m>f(x)</m> and <m>g(x)</m> be differentiable functions and let <m>h(x) = f(x) + g(x)</m>.
Then by the limit definition of the derivative,
</p>
</proof>
Let \(f(x)\) and \(g(x)\) be differentiable functions and let \(h(x) = f(x) + g(x)\text{.}\) Then by the limit definition of the derivative,
Example 36.3.
View Source for example
<example workspace="1in">
<statement>
<p>
Find the derivative of <m>f(x) = x^5 + e^x</m>.
</p>
</statement>
</example>
Find the derivative of \(f(x) = x^5 + e^x\text{.}\)
Example 36.4.
View Source for example
<example workspace="1in">
<statement>
<p>
Find the derivative of <m>f(x) = \sqrt{x} + x^3 + 7x</m>.
</p>
</statement>
</example>
Find the derivative of \(f(x) = \sqrt{x} + x^3 + 7x\text{.}\)
Example 36.5.
View Source for example
<example workspace="1in">
<statement>
<p>
Find the derivative of <m>f(x) = 5x^4</m>.
</p>
</statement>
</example>
Find the derivative of \(f(x) = 5x^4\text{.}\)
