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Derivative Functions

The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. We can formally define a derivative function as follows.

Definition

Let \(f\) be a function. The derivative function, denoted by \(f^{'} ,\) is the function whose domain consists of those values of \(x\) such that the following limit exists:

\[f^{'} \left(\right. x \left.\right) = \underset{h \rightarrow 0}{\text{lim}} \frac{f \left(\right. x + h \left.\right) - f \left(\right. x \left.\right)}{h} .\]

A function \(f \left(\right. x \left.\right)\) is said to be differentiable at \(a\) if \(f^{'} \left(\right. a \left.\right)\) exists. More generally, a function is said to be differentiable on \(S\) if it is differentiable at every point in an open set \(S ,\) and a differentiable function is one in which \(f^{'} \left(\right. x \left.\right)\) exists on its domain.

In the next few examples we use Equation 3.9 to find the derivative of a function.

Example 3.11

Finding the Derivative of a Square-Root Function

Find the derivative of \(f \left(\right. x \left.\right) = \sqrt{x} .\)

Solution

Start directly with the definition of the derivative function. Use Equation 3.1.

\[f^{'} \left(\right. x \left.\right) & = \underset{h \rightarrow 0}{\text{lim}} \frac{\sqrt{x + h} - \sqrt{x}}{h} & & & \begin{matrix}\text{Substitute} f \left(\right. x + h \left.\right) = \sqrt{x + h} \text{and} f \left(\right. x \left.\right) = \sqrt{x} \\ \text{into} f^{'} \left(\right. x \left.\right) = \underset{h \rightarrow 0}{\text{lim}} \frac{f \left(\right. x + h \left.\right) - f \left(\right. x \left.\right)}{h} .\end{matrix} \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{\sqrt{x + h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} & & & \begin{matrix}\text{Multiply numerator and denominator by} \\ \sqrt{x + h} + \sqrt{x} \text{without distributing in the} \\ \text{denominator}.\end{matrix} \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{h}{h \left(\right. \sqrt{x + h} + \sqrt{x} \left.\right)} & & & \text{Multiply the numerators and simplify}. \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{1}{\left(\right. \sqrt{x + h} + \sqrt{x} \left.\right)} & & & \text{Cancel the} h . \\ & = \frac{1}{2 \sqrt{x}} & & & \text{Evaluate the limit}.\]

Example 3.12

Finding the Derivative of a Quadratic Function

Find the derivative of the function \(f \left(\right. x \left.\right) = x^{2} - 2 x .\)

Solution

Follow the same procedure here, but without having to multiply by the conjugate.

\[f^{'} \left(\right. x \left.\right) & = \underset{h \rightarrow 0}{\text{lim}} \frac{\left(\right. \left(\left(\right. x + h \left.\right)\right)^{2} - 2 \left(\right. x + h \left.\right) \left.\right) - \left(\right. x^{2} - 2 x \left.\right)}{h} & & & \begin{matrix}\text{Substitute} f \left(\right. x + h \left.\right) = \left(\left(\right. x + h \left.\right)\right)^{2} - 2 \left(\right. x + h \left.\right) \text{and} \\ f \left(\right. x \left.\right) = x^{2} - 2 x \text{into} \\ f^{'} \left(\right. x \left.\right) = \underset{h \rightarrow 0}{\text{lim}} \frac{f \left(\right. x + h \left.\right) - f \left(\right. x \left.\right)}{h} .\end{matrix} \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{x^{2} + 2 x h + h^{2} - 2 x - 2 h - x^{2} + 2 x}{h} & & & \text{Expand} \left(\left(\right. x + h \left.\right)\right)^{2} - 2 \left(\right. x + h \left.\right) . \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{2 x h - 2 h + h^{2}}{h} & & & \text{Simplify}. \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{h \left(\right. 2 x - 2 + h \left.\right)}{h} & & & \text{Factor out} h \text{from the numerator}. \\ & = \underset{h \rightarrow 0}{\text{lim}} \left(\right. 2 x - 2 + h \left.\right) & & & \text{Cancel the common factor of} h . \\ & = 2 x - 2 & & & \text{Evaluate the limit}.\]

Checkpoint 3.6

Find the derivative of \(f \left(\right. x \left.\right) = x^{2} .\)

We use a variety of different notations to express the derivative of a function. In Example 3.12 we showed that if \(f \left(\right. x \left.\right) = x^{2} - 2 x ,\) then \(f^{'} \left(\right. x \left.\right) = 2 x - 2 .\) If we had expressed this function in the form \(y = x^{2} - 2 x ,\) we could have expressed the derivative as \(y^{'} = 2 x - 2\) or \(\frac{d y}{d x} = 2 x - 2 .\) We could have conveyed the same information by writing \(\frac{d}{d x} \left(\right. x^{2} - 2 x \left.\right) = 2 x - 2 .\) Thus, for the function \(y = f \left(\right. x \left.\right) ,\) each of the following notations represents the derivative of \(f \left(\right. x \left.\right) :\)

\[f^{'} \left(\right. x \left.\right) , \frac{d y}{d x} , y^{'} , \frac{d}{d x} \left(\right. f \left(\right. x \left.\right) \left.\right) .\]

In place of \(f^{'} \left(\right. a \left.\right)\) we may also use \(\begin{aligned} \frac{d y}{d x} \left|\right. \\ _{x = a} \end{aligned}\) Use of the \(\frac{d y}{d x}\) notation (called Leibniz notation) is quite common in engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form \(\frac{\Delta y}{\Delta x}\) where \(\Delta y\) is the difference in the \(y\) values corresponding to the difference in the \(x\) values, which are expressed as \(\Delta x\) (Figure 3.11). Thus the derivative, which can be thought of as the instantaneous rate of change of \(y\) with respect to \(x ,\) is expressed as

\[\frac{d y}{d x} = \underset{\Delta x \rightarrow 0}{\text{lim}} \frac{\Delta y}{\Delta x} .\]
The function y = f(x) is graphed and it shows up as a curve in the first quadrant. The x-axis is marked with 0, a, and a + Δx. The y-axis is marked with 0, f(a), and f(a) + Δy. There is a straight line crossing y = f(x) at (a, f(a)) and (a + Δx, f(a) + Δy). From the point (a, f(a)), a horizontal line is drawn; from the point (a + Δx, f(a) + Δy), a vertical line is drawn. The distance from (a, f(a)) to (a + Δx, f(a)) is denoted Δx; the distance from (a + Δx, f(a) + Δy) to (a + Δx, f(a)) is denoted Δy.
Figure 3.11 The derivative is expressed as \(\frac{d y}{d x} = \underset{\Delta x \rightarrow 0}{\text{lim}} \frac{\Delta y}{\Delta x} .\)

This lesson is part of:

Derivatives

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