### 4 Evaluating a derivative

The need to find the rate of change of a function at a particular point occurs often. We do this by finding the derivative of the function, and then evaluating the derivative at that point. When taking derivatives of trigonometric functions, any angles
**
must
**
be measured in radians. Consider a function,
$y\left(x\right)$
. We use the notation
$\frac{dy}{dx}\left(a\right)$
or
${y}^{\prime}\left(a\right)$
to denote the derivative of
$y$
evaluated at
$x=a$
. So
${y}^{\prime}\left(0.5\right)$
means the value of the derivative of
$y$
when
$x=0.5$
.

##### Example 6

Find the value of the derivative of $y={x}^{3}$ where $x=2$ . Interpret your result.

##### Solution

We have $y={x}^{3}$ and so $\frac{dy}{dx}=3{x}^{2}$ .

When $x=2$ , $\frac{dy}{dx}=3{\left(2\right)}^{2}=12$ , that is, $\frac{dy}{dx}\left(2\right)=12$ (Equivalently, ${y}^{\prime}\left(2\right)=12$ ).

The derivative is positive when $x=2$ and so $y$ is increasing at this point. When $x=2$ , $y$ is increasing at a rate of 12 vertical units per horizontal unit.