2.5: Introduction to the derivative

Learning Objectives

1. Explain the first examples of calculation of the derivative.
2. Describe how the derivative is obtained from an average rate of change.
3. Compute the derivative of very simple functions such as \(y=x^{2}\), and \(y=\) \(A x+B\).

We are ready for the the definition of the derivative.

Definition: 2.7 (The derivative)

The derivative of a function \(y=f(x)\) at a point \(x_{0}\) is the same as the instantaneous rate of change of \(f\) at \(x_{0}\). It is denoted \(\left.\frac{d y}{d x}\right|_{x_{0}}\) or \(f^{\prime}\left(x_{0}\right)\) and defined as

\[\left.\frac{d y}{d x}\right|_{x_{0}}=f^{\prime}\left(x_{0}\right)=\lim _{h \rightarrow 0} \frac{\left[f\left(x_{0}+h\right)-f\left(x_{0}\right)\right]}{h} . \nonumber \]

We can use this to update our definition of instantaneous velocity:

Definition: 2.8 (Velocity)

If \(y=f(t)\) is the position of an object at time \(t\) then the derivative \(f^{\prime}(t)\) at time \(t_{0}\) is the instantaneous velocity, also simply called the velocity of the object at that time.

As \(h \rightarrow 0\), the secant line approaches a tangent line. Use the slider for \(h\) to show this trend, and note that the slope of the secant line (average velocity) approaches the slope of the tangent line (instantaneous velocity) at the point \(x_{0}\).

Example 2.9 (Fomal calculation of velocity)

Use Gallileo’s formula to set up and calculate the derivative of Equation (2.1), and show that it corresponds to the instantaneous velocity obtained in Example 2.8.

Solution

We set up the calculation using limit notation, recalling Gallileo’s formula states \(y(t)=c t^{2}\). We compute

\[\begin{aligned} v\left(t_{0}\right) & =\lim _{h \rightarrow 0} \frac{y\left(t_{0}+h\right)-y\left(t_{0}\right)}{h} \\ & =\lim _{h \rightarrow 0} \frac{c\left(t_{0}+h\right)^{2}-c\left(t_{0}\right)^{2}}{h} \\ & =\lim _{h \rightarrow 0} c\left(\frac{\left(t_{0}^{2}+2 h t_{0}+h^{2}\right)-\left(t_{0}^{2}\right)}{h}\right) \\ & =\lim _{h \rightarrow 0} c\left(\frac{2 h t_{0}+h^{2}}{h}\right)=\lim _{h \rightarrow 0} c\left(2 t_{0}+h\right)=2 c t_{0} . \end{aligned} \nonumber \]

All steps but the last are similar to the calculation (and algebraic simplification) of average velocity (compare with Example 2.6). In the last step, we formally allow the time increment \(h\) to shrink, which is equivalent to taking \(\lim _{h \rightarrow 0}\).

Example 2.10

(Calculating the derivative of a function) Compute the derivative of the function \(f(x)=C x^{2}\) at some point \(x=x_{0}\).

Solution

In the previous example, we calculated the derivative of the function \(y=f(t)=c t^{2}\) with respect to \(t\). Here we merely have a similar (quadratic) function of \(x\). Thus, we have already solved this problem. By switching notation \(\left(t_{0} \rightarrow x_{0}\right.\) and \(c \rightarrow C\) ) we can write down the answer, \(2 C x_{0}\) at once. However, as practice, we rewrite the steps in the case of the general point \(x\)

For \(y=f(x)=C x^{2}\) we have

\[\begin{aligned} \frac{d y}{d x} & =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ & =\lim _{h \rightarrow 0} \frac{C(x+h)^{2}-C x^{2}}{h} \\ & =\lim _{h \rightarrow 0} C \frac{\left(x^{2}+2 x h+h^{2}\right)-x^{2}}{h} \\ & =\lim _{h \rightarrow 0} C \frac{\left(2 x h+h^{2}\right)}{h}=\lim _{h \rightarrow 0} C(2 x+h)=2 C x . \end{aligned} \nonumber \]

Evaluating this result for \(x=x_{0}\) we obtain the answer \(2 C x_{0}\).

We recognize from this definition that the derivative is obtained by starting with the slope of a secant line (average rate of change of \(f\) over the interval \(\left.x_{0} \leq x \leq x_{0}+h\right)\) and proceeds to shrink the interval \(\left(\lim _{h \rightarrow 0}\right)\) so that it approaches a single point \(\left(x_{0}\right)\). In later chapters, the resultant line is called the tangent line and the value obtained identified as the instantaneous rate of change of the function with respect to the variable \(x\) at the point of interest, \(x_{0}\). We explore properties and meanings of this concept in the next chapter.

Note: we have used different notations to denote the derivative of \(f(x)=y\). Further others exist. Each of the following may be used interchangeably:

\[f^{\prime}(x), \quad \frac{d f}{d x}, \quad \frac{d}{d x} f(x), \quad \frac{d y}{d x}, \quad y^{\prime}, \quad D f(x), \quad \text { and } \quad D y . \nonumber \]

These notations evolved for historical reasons and are used interchangeabley in science.