Definition 38.1. Definition of the Derivative.
The derivative of a function \(f(x)\) at any point \(x\) is defined as...
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Section 38 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 38.1 Derivative Rules
Rules for specific types of functions.
- Constant functions
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\(\displaystyle (k)' = 0\)
- Power functions
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\(\displaystyle (x^{n})' = nx^{n-1}\)
- Exponential functions
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\((a^{x})' = \ln(a) a^{x} \text{ (for } a>0\text{)}\)\((e^{x})' = e^{x}\)
- Logarithmic functions
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\(\displaystyle (\ln(x))' = \frac{1}{x}\)
- Trigonometric functions
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\((\sin(x))' = \cos(x)\text{.}\)\((\cos(x))' = -\sin(x)\text{.}\)\((\tan(x))' = \sec^{2}(x) = \frac{1}{\cos^{2}(x)}\text{.}\)\((\arcsin(x))' = \frac{1}{\sqrt{1-x^{2}}}\text{.}\)\((\arctan(x))' = \frac{1}{1+x^{2}}\text{.}\)
Rules for combinations of functions.
- Constant multiples
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\(\displaystyle (k\cdot f(x))' = k\cdot f'(x)\)
- Sum and difference
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\(\displaystyle (f(x) \pm g(x))' = f'(x) \pm g'(x)\)
- Products (the product rule)
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\(\displaystyle (f(x)\cdot g(x))' = f'(x)g(x) + f(x)g'(x)\)
- Quotients (the quotient rule)
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\(\displaystyle \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^{2}}\)
- Compositions (the chain rule)
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\(\displaystyle (f(g(x)))' = f'(g(x))\cdot g'(x)\)
Handout 38.2 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 38.2.
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 38.3.
Find the derivative of \(f(x) = x^5 + e^x\text{.}\)
Example 38.4.
Find the derivative of \(f(x) = \sqrt{x} + x^3 + 7x\text{.}\)
Example 38.5.
Find the derivative of \(f(x) = 5x^4\text{.}\)
Handout 38.3 A handout with a long list
It is common for a handout, especially a set of lecture notes, to consist of a long list of bullets, possibly more than would fit on a page. This handout should test that such a handout can have its items split between pages even if they belong to the same paragraph.
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This is the first item in a long list. It should be on the first page of the handout.
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This is the second item in a long list. It should be on the first page of the handout.
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This is the third item in a long list. It should be on the second page of the handout.Maybe we have a second paragraph in the same list item. This is really just to help fill up space. Maybe I should have used some of that standard space filler, but I’m just typing now because I can’t remember what is call.
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I suppose I could just start copy/pasting now.
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur.Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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A late footnote -
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
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See, I figured it out!
And that’s the end of the list. Did this go onto a second page?
