Activity 23.
Use only the rules for constant, power, and exponential functions, together with the Constant Multiple and Sum Rules, to compute the derivative of each function below with respect to the given independent variable. Note well that we do not yet know any rules for how to differentiate the product or quotient of functions. This means that you may have to do some algebra first on certain functions below before you can actually use existing rules to compute the desired derivative formula. For each function whose derivative you find, label the derivative with its name using proper notation such as \(f'(x)\text{,}\) \(h'(z)\text{,}\) \(dr/dt\text{,}\) etc.
(a)
\(f(x) = x^{5/3} - x^4 + 2^x\)
Hint.
Use the sum rule.
Answer.
\(f'(x) = \frac{5}{3}x^{2/3} - 4 x^3 + 2^x \ln(2)\text{.}\)
Solution.
\(f(x) = x^{5/3} - x^4 + 2^x\text{,}\) so by the sum rule, \(f'(x) = \frac{5}{3}x^{2/3} - 4 x^3 + 2^x \ln(2)\text{.}\)
(b)
\(g(x) = 14e^x + 3x^5 - x\)
Hint.
Use the sum rule together with the constant multiple rule.
Answer.
\(g'(x) = 14e^x + 3 \cdot 5x^4 - 1\text{.}\)
Solution.
\(g(x) = 14e^x + 3x^5 - x\text{,}\) so by the sum and constant multiple rules, \(g'(x) = 14e^x + 3 \cdot 5x^4 - 1\text{.}\)
(c)
\(h(z) = \sqrt{z} + \frac{1}{z^4} + 5^z\)
Hint.
How can you rewrite \(\sqrt{z}\) using exponents?
Answer.
\(h'(z) = \frac{1}{2}z^{-1/2} - 4z^{-5} + 5^z \ln(5)\text{.}\)
Solution.
\(h(z) = \sqrt{z} + \frac{1}{z^4} + 5^z = z^{1/2} + z^{-4} + 5^z\text{,}\) thus \(h'(z) = \frac{1}{2}z^{-1/2} - 4z^{-5} + 5^z \ln(5)\text{.}\)
(d)
\(r(t) = \sqrt{53} \, t^7 - \pi e^t + e^4\)
Hint.
Is \(e^4\) a constant or variable?
Answer.
\(\frac{dr}{dt} = \sqrt{53} \cdot 7 t^6 - \pi e^t\text{.}\)
Solution.
Since \(r(t) = \sqrt{53} \, t^7 - \pi e^t + e^4\) and \(\sqrt{53}\text{,}\) \(\pi\text{,}\) and \(e^4\) are all constants, it follows from the sum and constant multiple rules, as well as the derivative of a constant rule, that \(\frac{dr}{dt} = \sqrt{53} \cdot 7 t^6 - \pi e^t\text{.}\) (Note particularly that \(\frac{d}{dt}[e^4] = 0\) since \(e^4\) is constant.
(e)
\(s(y) = (y^2 + 1)(y^2 - 1)\)
Hint.
Expand the product before attempting to find the derivative.
Answer.
\(\frac{ds}{dy} = 4y^3\text{.}\)
Solution.
\(s(y) = (y^2 + 1)(y^2 - 1)= y^4 - 1\text{,}\) thus \(\frac{ds}{dy} = 4y^3\text{.}\)
(f)
\(q(x) = \frac{x^3 - x + 2}{x}\)
Hint.
Rewrite the single fraction as a sum of three fractions, and simplify.
Answer.
\(q'(x) = 2x - 2x^{-2}\text{.}\)
Solution.
\(q(x) = \frac{x^3 - x + 2}{x} = \frac{x^3}{x} - \frac{x}{x} + \frac{2}{x} = x^2 - 1 + 2x^{-1}\text{.}\) Now it follows that \(q'(x) = 2x - 2x^{-2}\text{.}\)
(g)
\(p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12\)
Hint.
Note that “\(a\)” is the independent variable.
Answer.
\(p'(a) = 12a^3 - 6 a^2 + 14a - 1\text{.}\)
Solution.
\(p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12\text{,}\) so \(p'(a) = 12a^3 - 6 a^2 + 14a - 1\text{.}\)
