Activity 2.1.3

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 $f'(x)$, $h'(z)$, $dr/dt$, etc.

1. $f(x) = x^{5/3} - x^4 + 2^x$

2. $g(x) = 14e^x + 3x^5 - x$

3. $h(z) = \sqrt{z} + \frac{1}{z^4} + 5^z$

4. $r(t) = \sqrt{53} \, t^7 - \pi e^t + e^4$

5. $s(y) = (y^2 + 1)(y^2 - 1)$

6. $q(x) = \frac{x^3 - x + 2}{x}$

7. $p(a) = 3a^4 - 2a^3 + 7a^2 - a + 12$