Activity 22.
Use the rules for constant, power, and exponential functions to determine the derivative of each of the following functions. For each, state your answer using full and proper notation, labeling the derivative with its name. For example, if you are given a function \(h(z)\text{,}\) you should use notation such as “\(h'(z) =\)” or “\(\frac{dh}{dz} =\)” as part of your response.
(a)
\(f(t) = \pi\)
Hint.
Is \(\pi\) a variable or a constant?
Answer.
\(f'(t) = 0\text{.}\)
Solution.
\(f(t) = \pi\) is constant, so \(f'(t) = 0\text{.}\)
(b)
\(g(z) = 7^z\)
Hint.
Is \(g\) a power or exponential function?
Answer.
\(g'(z) = 7^z \ln(7)\text{.}\)
Solution.
\(g(z) = 7^z\) is an exponential function, so \(g'(z) = 7^z \ln(7)\text{.}\)
(c)
\(h(w) = w^{3/4}\)
Hint.
Is \(h\) a power or exponential function?
Answer.
\(h'(w) = \frac{3}{4} w^{-1/4}\text{.}\)
Solution.
\(h(w) = w^{3/4}\) is a power function, thus \(h'(w) = \frac{3}{4} w^{-1/4}\text{.}\)
(d)
\(p(x) = 3^{1/2}\)
Hint.
Is \(3^{1/2}\) a constant or a variable?
Answer.
\(\frac{dp}{dx} = 0\text{.}\)
Solution.
\(p(x) = 3^{1/2}\) is constant, and therefore \(\frac{dp}{dx} = 0\text{.}\)
(e)
\(r(t) = (\sqrt{2})^t\)
Hint.
\(\sqrt{2}\) is a constant
Answer.
\(r'(t) = (\sqrt{2})^t \ln (\sqrt{2})\text{.}\)
Solution.
\(r(t) = (\sqrt{2})^t\) is exponential (since \(\sqrt{2}\) is a constant), and so we have \(r'(t) = (\sqrt{2})^t \ln (\sqrt{2})\text{.}\)
(f)
\(s(q) = q^{-1}\)
Hint.
Remember the notation here means “take the derivative with respect to \(q\) of \(q^{-1}\text{.}\)”
Answer.
\(\frac{d}{dq}[q^{-1}] = -q^{-2}\text{.}\)
Solution.
\(\frac{d}{dq}[q^{-1}] = -q^{-2}\text{,}\) by the rule for power functions.
(g)
\(m(t) = \frac{1}{t^3}\)
Hint.
Rewrite the fraction using a negative exponent.
Answer.
\(\frac{dm}{dt} = -3t^{-4} = -\frac{3}{t^4}\text{.}\)
Solution.
\(m(t) = \frac{1}{t^3} = t^{-3}\text{,}\) so \(\frac{dm}{dt} = -3t^{-4} = -\frac{3}{t^4}\text{.}\)
