12.4 Derivatives  (Page 2/18)

Understanding the instantaneous rate of change

Now that we can find the average rate of change, suppose we make $h$ in [link] smaller and smaller. Then $a+h$ will approach $a$ as $h$ gets smaller, getting closer and closer to 0. Likewise, the second point $\left(a+h,f(a+h)\right)$ will approach the first point, $\left(a,f(a)\right).$ As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at $x=a,$ and the slope of the secant line will get closer and closer to the slope of the tangent at $x=a.$ See [link] .

Because we are looking for the slope of the tangent at $x=a,$ we can think of the measure of the slope of the curve of a function $f$ at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change , or the derivative of the function at $x=a.$ Both can be found by finding the limit of the slope of a line connecting the point at $x=a$ with a second point infinitesimally close along the curve. For a function $f$ both the instantaneous rate of change of the function and the derivative of the function at $x=a$ are written as $f\text{'}(a),$ and we can define them as a two-sided limit    that has the same value whether approached from the left or the right.

The expression by which the limit is found is known as the difference quotient .

Derivatives: interpretations and notation

The derivative    of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.

• The derivative of a function $f(x)$ at a point $x=a$ is the slope of the tangent line to the curve $f(x)$ at $x=a.$ The derivative of $f(x)$ at $x=a$ is written $f^{\prime }(a).$
• The derivative $f^{\prime }(a)$ measures how the curve changes at the point $\left(a,f(a)\right).$
• The derivative $f^{\prime }(a)$ may be thought of as the instantaneous rate of change of the function $f(x)$ at $x=a.$
• If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time $t=a.$