Derivatives of special functions [3.2,5]

Derivatives of polynomials [3.1]

Between class on Wednesday and lab on Thursday, we got to these three sets of derivative rules:

Constant * function: if $c$ is a constant, then $$\frac{d}{dx}\left[ c f(x)\right]= c f'(x).$$

Sum and difference: $$\frac{d}{dx}\left[ f(x)+g(x)\right]=f'(x)+g'(x).$$ $$\frac{d}{dx}\left[ f(x)-g(x)\right]=f'(x)-g'(x).$$

Power rule: if $n$ is a constant, then $$\frac{d}{dx}\left[ x^n \right]=n x^{n-1}.$$

Example: Using *only* these rules I can take derivatives of things like $y(t)=\frac{4+t}{5t^2} $: $$\begineq \frac{d}{dt}\left[ \frac{4+t}{5t^2} \right] =& \frac{d}{dt}\left[ \frac{4}{5t^2}+\frac{t}{5t^2} \right] = \frac{d}{dt}\left[ \frac{4}{5}t^{-2}+\frac{1}{5}t^{-1} \right]\\ =& \frac{4}{5}(-2)t^{-3}+\frac 15 (-1)t^{-2} =-\frac{8}{5}\frac{1}{t^{3}}-\frac 15 \frac{1}{t^{2}}\\ =&-\frac{8}{5t^3}-\frac{1}{5t^2}=-\frac{8+t}{5t^3} \endeq $$

Exponential: $a^x$

In the lab, several of you showed, or got close to the point of showing this one:

The story so far... $$\begineq f'(x)=&\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}=\lim_{h\to 0} \frac{a^{(x+h)}-a^x}{h}\\ =&\lim_{h\to 0} \frac{a^xa^h-a^x}{h} =\lim_{h\to 0} a^x\frac{a^h-1}{h}\\ =&a^x\color{blue}\lim_{h\to 0}\frac{a^h-1}{h}\\ \endeq $$ Here's a graph of $g(a)=\frac{a^{0.03}-1}{0.03}$:

What function could this be? $\sin(a), \cos(a), e^a, \ln(a), 1/a, \sqrt a...$

Pin down your guess by making $h=0.03$ even smaller...

Exponential function: if $a$ is a constant, then $$\frac{d}{dx}\left[ a^x \right]= (\ln a)a^x.$$

Examples

$$\frac{d}{dx}e^x=?$$

$$\frac{d}{dx}10^x=?$$

$$\frac{d}{dx}e^{kx}=?$$ show / hide

In CoCalc,

• log(x) means $\ln x\equiv \log_{e}x$, which you can also write as ln(x).
• log(x,10) means $\log_{10} x$.

• Showing the other special functions...

Natural log

Here is a graph of $\color{red}{f(x)=\ln x}$, and the difference quotient (not the ldq) $\color{green}{\frac{f(x+h)-f(x)}{h}}$ (where you can slide $h$).

What function does the difference quotient look like??

Natural log: $$\frac{d}{dx}\left[ \ln x \right]=\frac 1x. $$

Sine and cosine

Here is a graph of $\color{red}{f(x)=\sin x}$, and the difference quotient (not the ldq) $\color{green}{\frac{f(x+h)-f(x)}{h}}$ (where you can slide $h$).

What function could this be?: $\sin(x), \cos(x), e^x, \ln(x), 1/x, \sqrt x...$

Sine of $x$: $$\frac{d}{dx}\left[ \sin x \right] =\cos x. $$ Cosine of $x$: $$\frac{d}{dx}\left[ \cos x \right]=-\sin x. $$

Example

:

Examine the graph of $\sin x$ and sketch a graph of its concavity (second derivative). (Where is it positive? negative? zero?).

Now, calculate the second derivative of $\sin x$...
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