Elementary derivative rules

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 $h(z)$, you should use notation such as “$h'(z) =$” or “$\frac{dh}{dz} =$” as part of your response.

1. $f(t) = \pi$

2. $g(z) = 7^z$

3. $h(w) = w^{3/4}$

4. $p(x) = 3^{1/2}$

5. $r(t) = (\sqrt{2})^t$

6. $s(q) = q^{-1}$

7. $m(t) = \frac{1}{t^3}$

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 bel…

Each of the following questions asks you to use derivatives to answer key questions about functions. Be sure to think carefully about each question and to use proper notation in your responses.

1. Find the slope of the tangent line to $h(z) = \sqrt{z} + \frac{1}{z}$ at the point where $z = 4$.

2. A population of cells is growing in such a way that its total number in millions is given by the function $P(t) = 2(1.37)^t + 32$, where $t$ is measured in days.

   1. Determine…

Let $f$ and $g$ be differentiable functions for which the following information is known: $f(2) = 5$, $g(2) = -3$, $f'(2) = -1/2$, $g'(2) = 2$.

1. Let $h$ be the new function defined by the rule $h(x) = 3f(x) - 4g(x)$. Determine $h(2)$ and $h'(2)$.

2. Find an equation for the tangent line to $y = h(x)$ at the point $(2,h(2))$.

3. Let $p$ be the function defined by the rule $p(x) = -2f(x) + \frac{1}{2}g(x)$. Is $p$ increasing, decreasing, or neither at $a = 2$? Why?

4. Estimate the value of $p(2.03)$ by using the local linearization of $p$ at the point $(2,p(2))$.

Consider the functions $r(t) = t^t$ and $s(t) = \arccos(t)$, for which you are given the facts that $r'(t) = t^t(\ln(t) + 1)$ and $s'(t) = -\frac{1}{\sqrt{1-t^2}}$. Do not be concerned with where these derivative formulas come from. We restrict our interest in both functions to the domain $0 \lt t \lt 1$.

1. Let $w(t) = 3t^t - 2\arccos(t)$. Determine $w'(t)$.

2. Find an equation for the tangent line to $y = w(t)$ at the point $(\frac{1}{2}, w(\frac{1}{2}))$.

3. Let $v(t) = t^t + \arccos(t)$. Is $v$ increasing or decreasing at the instant $t = \frac{1}{2}$? Why?

Let $f(x) = a^x$. The goal of this problem is to explore how the value of $a$ affects the derivative of $f(x)$, without assuming we know the rule for $\frac{d}{dx}[a^x]$ that we have stated and used in earlier work in this section.

1. Use the limit definition of the derivative to show that .

2. Explain why it is also true that .

3. Use computing technology and small values of $h$ to estimate the value of when $a = 2$. Do likewise when $a = 3$.

4. Note that it w…