§2.1 Elementary derivative rules

Introduction

Some Key Notation

Constant, Power, and Exponential Functions

Functions of the form $f(x) = x^n$, where $n = 1, 2, 3, \ldots$, are often called power functions. The first two questions below revisit work we did earlier in Chapter 1, and the following questions extend those ideas to higher powers of $x$.

Use the limit definition of the derivative to find $f'(x)$ for $f(x) = x^2$.

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!

$f(t) = \pi$

$g(z) = 7^z$

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

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

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

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

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