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Chapter 7: Problem 31

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

Short Answer

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

Step by step solution

01

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 \( f(x) \).
02

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: \( f'(x) = \lim_{h \to 0} \frac{(x + h)^{2} - 4 - (x^{2} - 4)}{h} \).
03

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 \( 2x \).
04

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 \( 2x \).

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Most popular questions from this chapter

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

Identify the inside function, \(u=g(x)\), and the outside function, \(y=f(u)\). $$ y=f(g(x)) \quad u=g(x) \quad y=f(u) $$ $$ y=\sqrt{1-x^{2}} $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(25+x^{2}\right)^{-1 / 2} $$

Use the given information to find \(f^{\prime}(2)\) \(g(2)=3\) and \(g^{\prime}(2)=-2\) \(h(2)=-1 \quad\) and \(\quad h^{\prime}(2)=4\) $$ f(x)=2 g(x)+h(x) $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\sqrt{x^{2}-2 x+1} $$
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