Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ f(x)=1-x^{2} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 1 - x^{2}\) is \(f'(x)=-2x\).

Step 1: Insert f(x+h) into the limit definition

First, replace the \(x\) in \(f(x)\) with \(x+h\). This results in \(f(x+h) = 1 - (x+h)^{2}\). Now, the limit definition is \(f'(x)= \lim _{h \rightarrow 0} \frac{1 - (x+h)^{2}- (1 - x^{2})}{h}\).

Step 2: Simplify the numerator

Next, simplify the numerator of the fraction inside the limit. The \(1\) and \(-1\) cancel each other out, and expanding \(-(x+h)^{2}\) gives \(-x^{2}-2xh-h^{2}\). This leads to \( f'(x) = \lim _{h \rightarrow 0} \frac{-x^{2}-2xh-h^{2}+x^{2}}{h}\). This further simplifies to \(f'(x)= \lim _{h \rightarrow 0}\frac{-2xh - h^{2}}{h}\).

Step 3: Cancel out h

In this step, cancel out \(h\) from the numerator and the denominator, resulting in \(f'(x)= \lim _{h \rightarrow 0}(-2x - h)\).

Step 4: Take the limit

Lastly, take the limit as \(h\) approaches 0, remaining with \(f'(x)=-2x\).