Activity 32.
For each function given below, find its derivative.
(a)
\(h(x) = x^2\ln(x)\)
Hint.
Is \(h\) a product, quotient, or composition of basic functions?
Answer.
\(h'(x) = x + 2x\ln(x)\text{.}\)
Solution.
By the product rule,
\begin{equation*}
h'(x) = x^2\cdot \frac{1}{x} + \ln(x) \cdot 2x = x + 2x\ln(x)\text{.}
\end{equation*}
(b)
\(p(t) = \frac{\ln(t)}{e^t + 1}\)
Hint.
Is \(p\) a product, quotient, or composition of basic functions?
Answer.
\(p'(t) = \frac{(e^t + 1) \frac{1}{t} - \ln(t) \cdot e^t}{(e^t + 1)^2}\text{.}\)
Solution.
By the quotient rule,
\begin{equation*}
p'(t) = \frac{(e^t + 1) \frac{1}{t} - \ln(t) \cdot e^t}{(e^t + 1)^2}\text{.}
\end{equation*}
(c)
\(s(y) = \ln(\cos(y) + 2)\)
Hint.
Is \(s\) a product, quotient, or composition of basic functions?
Answer.
\(s'(y) = \frac{1}{\cos(y) + 2} \cdot (-\sin(y))\text{.}\)
Solution.
The chain rule tells us that
\begin{equation*}
s'(y) = \frac{1}{\cos(y) + 2} \cdot (-\sin(y))\text{.}
\end{equation*}
(d)
\(z(x) = \tan(\ln(x))\)
Hint.
Is \(z\) a product, quotient, or composition of basic functions?
Answer.
\(z'(x) = \sec^2(\ln(x)) \cdot \frac{1}{x}\text{.}\)
Solution.
Again using the chain rule,
\begin{equation*}
z'(x) = \sec^2(\ln(x)) \cdot \frac{1}{x}\text{.}
\end{equation*}
(e)
\(m(z) = \ln(\ln(z))\)
Hint.
Is \(m\) a product, quotient, or composition of basic functions?
Answer.
\(m'(z) = \frac{1}{\ln(z)} \cdot \frac{1}{z}\text{.}\)
Solution.
Noting that \(m\) is composite with the natural logarithm function serving as both the inner and outer function, we find that
\begin{equation*}
m'(z) = \frac{1}{\ln(z)} \cdot \frac{1}{z}\text{.}
\end{equation*}
