Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ f(x)=x^{2}-4 $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = x^{2} - 4 \) by using the limit definition is \( f'(x) = 2x \).

Step 1: Write down the function and the definition of derivative

Given is the function \( f(x) = x^{2} - 4 \). The limit definition of the derivative is \( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \). We will be using this definition to find the derivative of \( f(x) \).

Step 2: Substitute \( f(x) \) and \( f(x + h) \) into the definition

To apply this definition, we must evaluate \( f(x + h) \) and \( f(x) \). For our function, \( f(x+h) = (x + h)^{2} - 4 \) and \( f(x) = x^{2} - 4 \). Substituting these into the definition gives us: \( f'(x) = \lim_{h \to 0} \frac{(x + h)^{2} - 4 - (x^{2} - 4)}{h} \).

Step 3: Simplify the expression

We now simplify the expression in the numerator. The 4 from \( -4 \) cancels out, leaving us with: \( \lim_{h \to 0} \frac{(x + h)^{2} - x^{2}}{h} \). This further simplifies to \( \lim_{h \to 0} \frac{x^{2} + 2hx + h^{2} - x^{2}}{h} \), or \( \lim_{h \to 0} \frac{2hx + h^{2}}{h} \). Given \( h \neq 0 \), one can write this as \( 2x + h \). As \( h \) tends to 0, the expression simplifies to \( 2x \).

Step 4: State the result

The derivative of the function \( f(x) = x^{2} - 4 \) using the limit definition is \( f'(x) = 2x \). This result means that for any point on the curve \( f(x) = x^{2} - 4 \), the slope of the tangent line to the curve at that point is given by \( 2x \).