The Derivative of a Function at a Point

The type of limit we compute in order to find the slope of the line tangent to a function at a point occurs in many applications across many disciplines. These applications include velocity and acceleration in physics, marginal profit functions in business, and growth rates in biology. This limit occurs so frequently that we give this value a special name: the derivative. The process of finding a derivative is called differentiation.

Definition

Let \(f \left(\right. x \left.\right)\) be a function defined in an open interval containing \(a .\) The derivative of the function \(f \left(\right. x \left.\right)\) at \(a ,\) denoted by \(f^{'} \left(\right. a \left.\right) ,\) is defined by

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

provided this limit exists.

Alternatively, we may also define the derivative of \(f \left(\right. x \left.\right)\) at \(a\) as

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

Example 3.4

Estimating a Derivative

For \(f \left(\right. x \left.\right) = x^{2} ,\) use a table to estimate \(f^{'} \left(\right. 3 \left.\right)\) using Equation 3.5.

Solution

Create a table using values of \(x\) just below \(3\) and just above \(3 .\)

\(x\)	\(\frac{x^{2} - 9}{x - 3}\)
\(2.9\)	\(5.9\)
\(2.99\)	\(5.99\)
\(2.999\)	\(5.999\)
\(3.001\)	\(6.001\)
\(3.01\)	\(6.01\)
\(3.1\)	\(6.1\)

After examining the table, we see that a good estimate is \(f^{'} \left(\right. 3 \left.\right) = 6.\)

Checkpoint 3.2

For \(f \left(\right. x \left.\right) = x^{2} ,\) use a table to estimate \(f^{'} \left(\right. 3 \left.\right)\) using Equation 3.6.

Example 3.5

Finding a Derivative

For \(f \left(\right. x \left.\right) = 3 x^{2} - 4 x + 1 ,\) find \(f^{'} \left(\right. 2 \left.\right)\) by using Equation 3.5.

Solution

Substitute the given function and value directly into the equation.

\[\begin{aligned} f^{'} \left(\right. 2 \left.\right) & = \underset{x \rightarrow 2}{\text{lim}} \frac{f \left(\right. x \left.\right) - f \left(\right. 2 \left.\right)}{x - 2} & & & \text{Apply the definition}. \\ & = \underset{x \rightarrow 2}{\text{lim}} \frac{\left(\right. 3 x^{2} - 4 x + 1 \left.\right) - 5}{x - 2} & & & \text{Substitute} f \left(\right. x \left.\right) = 3 x^{2} - 4 x + 1 \text{and} f \left(\right. 2 \left.\right) = 5 . \\ & = \underset{x \rightarrow 2}{\text{lim}} \frac{\left(\right. x - 2 \left.\right) \left(\right. 3 x + 2 \left.\right)}{x - 2} & & & \text{Simplify and factor the numerator}. \\ & = \underset{x \rightarrow 2}{\text{lim}} \left(\right. 3 x + 2 \left.\right) & & & \text{Cancel the common factor}. \\ & = 8 & & & \text{Evaluate the limit}. \end{aligned}\]

Example 3.6

Revisiting the Derivative

For \(f \left(\right. x \left.\right) = 3 x^{2} - 4 x + 1 ,\) find \(f^{'} \left(\right. 2 \left.\right)\) by using Equation 3.6.

Solution

Using this equation, we can substitute two values of the function into the equation, and we should get the same value as in Example 3.5.

\[f^{'} \left(\right. 2 \left.\right) & = \underset{h \rightarrow 0}{\text{lim}} \frac{f \left(\right. 2 + h \left.\right) - f \left(\right. 2 \left.\right)}{h} & & & \text{Apply the definition}. \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{\left(\right. 3 \left(\left(\right. 2 + h \left.\right)\right)^{2} - 4 \left(\right. 2 + h \left.\right) + 1 \left.\right) - 5}{h} & & & \begin{matrix}\text{Substitute} f \left(\right. 2 \left.\right) = 5 \text{and} \\ f \left(\right. 2 + h \left.\right) = 3 \left(\left(\right. 2 + h \left.\right)\right)^{2} - 4 \left(\right. 2 + h \left.\right) + 1 .\end{matrix} \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{3 h^{2} + 8 h}{h} & & & \text{Simplify the numerator}. \\ & = \underset{h \rightarrow 0}{\text{lim}} \frac{h \left(\right. 3 h + 8 \left.\right)}{h} & & & \text{Factor the numerator}. \\ & = \underset{h \rightarrow 0}{\text{lim}} \left(\right. 3 h + 8 \left.\right) & & & \text{Cancel the common factor}. \\ & = 8 & & & \text{Evaluate the limit}.\]

The results are the same whether we use Equation 3.5 or Equation 3.6.

Checkpoint 3.3

For \(f \left(\right. x \left.\right) = x^{2} + 3 x + 2 ,\) find \(f^{'} \left(\right. 1 \left.\right) .\)

This lesson is part of:

Derivatives

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