§2.1 Elementary derivative rules

Introduction

Some Key Notation

Constant, Power, and Exponential Functions

If $f(x) = 7$, then $f'(x) = 0$. Similarly, $\frac{d}{dx} [\sqrt{3}] = 0$.

Using the rule for power functions, we can compute the following derivatives. If $g(z) = z^{-3}$, then $g'(z) = -3z^{-4}$. Similarly, if $h(t) = t^{7/5}$, then $\frac{dh}{dt} = \frac{7}{5}t^{2/5}$, and $\frac{d}{dq} [q^{\pi}] = \pi q^{\pi - 1}$.

If $f(x) = 2^x$, then $f'(x) = 2^x \ln(2)$. Similarly, for $p(t) = 10^t$, $p'(t) = 10^t \ln(10)$. It is especially important to note that when $a = e$, where $e$ is the base of the natural logarithm function, we have that

since $\ln(e) = 1$. This is an extremely important property of the function $e^x$: its derivative function is itself!

Use the rule for power functions to differentiate $h(t) = t^{7/5}$. Which expression is $\frac{dh}{dt}$?

Let $g(z) = 7^z$. Which expression is $g'(z)$?

Which of the following functions has derivative identically equal to zero?

For $m(t) = \frac{1}{t^3}$, compute the instantaneous rate of change $m'(1)$.

For $h(w) = w^{3/4}$, compute $h'(1)$ as a fraction.

If $f(t) = \pi$, then $f'(t) = 0$ for every value of $t$.