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Handout 36.2 Guided Notes: Derivatives of sums
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.
The derivative of a function
\(f(x)\) at any point
\(x\) is defined as...
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.
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.
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.
Find the derivative of
\(f(x) = x^5 + e^x\text{.}\)
Example 36.4.
Find the derivative of
\(f(x) = \sqrt{x} + x^3 + 7x\text{.}\)
Example 36.5.
Find the derivative of
\(f(x) = 5x^4\text{.}\)
