Section 36 Handouts
About Handouts.
Like worksheets, a handout is a division that is inteded to be printed for use in a classroom. In HTML output, you get the same print preview and page layout as you do with worksheets; in PDF, these will start on a new page with possibly different margins than the rest of the document.
Unlike worksheets, handouts do not have a special class of exercises or activities (exercises in a handout are treated like an inline exercise). The other main distinction is that a handout lets workspace be specified on pretty much any block or paragraph element, not just exercises, tasks, and project-like elements. This allows them to be used in the creation of guided notes containing some premade content with lots of space for students to fill in details during class.
Handout 36.1 Derivative Rules
Rules for specific types of functions.
- Constant functions
- \(\displaystyle (k)' = 0\)
- Power functions
- \(\displaystyle (x^{n})' = nx^{n-1}\)
- Exponential functions
-
\((a^{x})' = \ln(a) a^{x} \text{ (for } a>0\text{)}\)\((e^{x})' = e^{x}\)
- Logarithmic functions
- \(\displaystyle (\ln(x))' = \frac{1}{x}\)
- Trigonometric functions
-
\((\sin(x))' = \cos(x)\text{.}\)\((\cos(x))' = -\sin(x)\text{.}\)\((\tan(x))' = \sec^{2}(x) = \frac{1}{\cos^{2}(x)}\text{.}\)\((\arcsin(x))' = \frac{1}{\sqrt{1-x^{2}}}\text{.}\)\((\arctan(x))' = \frac{1}{1+x^{2}}\text{.}\)
Rules for combinations of functions.
- Constant multiples
- \(\displaystyle (k\cdot f(x))' = k\cdot f'(x)\)
- Sum and difference
- \(\displaystyle (f(x) \pm g(x))' = f'(x) \pm g'(x)\)
- Products (the product rule)
- \(\displaystyle (f(x)\cdot g(x))' = f'(x)g(x) + f(x)g'(x)\)
- Quotients (the quotient rule)
- \(\displaystyle \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^{2}}\)
- Compositions (the chain rule)
- \(\displaystyle (f(g(x)))' = f'(g(x))\cdot g'(x)\)
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.
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 \(f(x) = 5x^4\text{.}\)
