$dx's$ is zero. The result, then, is $(x + dx)^{n} = x^{n} + nx^{n-1}dx$, which is the statement of the binomial theorem for $x$ plus a dibbl taken to any power $n$.\
Eq \[eq:deriv\] gives, for $y(x) = x^{n}$: $$\begin{aligned}
\frac{dy}{dx}& = &\frac{d(x^{n})}{dx} \nonumber \\
             & = &\frac{(x^{n} + nx^{n-1}dx) - x^{n}}{dx} \nonumber \\
             & = &\frac{nx^{n-1}dx}{dx} = nx^{n-1} \label{eq:power}\end{aligned}$$ Eq \[eq:power\] is often called “the power law”.

[**EXERCISE**]{}

[**2.7**]{} Calculate the derivatives $y' \equiv \frac{dy}{dx}$ ($\equiv$ means that the two expressions mean the same thing), using the method of Eq \[eq:power\] when\
(a) $y(x) = x^{9}$;\
(b) $y(x) = 5x^{17}$;\
(c) $y(x) = 6x^{5} + 4x^{4}$\
Show all of your work.

“DERIVATIVE” IS A FANCY NAME\
FOR VELOCITY OR RATE OF CHANGE
------------------------------

[**EXAMPLE 2.1**]{}

Find an expression for the straight line that is tangent to the curve described by $y(x) = \frac{1}{7}x^{5}$ at the the point $x = 2$. Show your work and graph the result.\
[**Solution:**]{}\
(1) Let $y_{t}(x) = a + bx$ be the equation for the straight line. Then our job is to determine the values of the constants $a$ and $b$.\
(2) Since $y_{t}(x)$ is a tangent, it must have the same slope as 