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Handout Monday 9/22
Last time we learned how to take derivatives of power functions, like \(f(x) = x^4\text{,}\) but also \(f(x) = \frac{1}{x^2} = x^{-2}\) and \(f(x) = \sqrt[3]{x} = x^{-1/3}\text{.}\) The rule is that for any constant \(n\text{,}\) if
\begin{equation*}
f(x) = x^n
\end{equation*}
then
\begin{equation*}
f'(x) = nx^{n-1}\text{.}
\end{equation*}
We also looked at how to combine functions: \(f(x) = g_1(x) +g_2(x)\) has derivative \(f'(x) = g_1'(x) + g_2'(x)\text{.}\) And if \(f(x)\) is a constant multiple of another function, say \(f(x) = 5g(x)\text{,}\) then \(f'(x) = 5g'(x)\text{.}\) For example, \(f(x) = 7x^3\) will have derivative \(f'(x) = 7\cdot(3x^2) = 21x^2\text{.}\)
Today, let’s practice these derivatives. Write down a bunch of functions that we can take derivatives of and do it. Include some that need to be simplified first.
What if we have \(f(x) = 2^x\text{?}\) Is the derivative \(f'(x) = x2^{x-1}\text{?}\) No! We can verify this by graphing both. So actually we do want to graph these exponential functions and numerically compute the derivatives. It turns out that for any constant \(a\text{,}\) the function \(f(x) = a^x\) has derivative \(f'(x) = a^x \ln(a)\text{.}\)
In other words, the derivative if an exponential function is proportional to itself, with constant of proportionality \(\ln(a)\) where \(a\) is the base. Note that since \(\ln(e) = 1\text{,}\) this says that \(\frac{d}{dx}e^x = e^x\text{.}\)
