Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ f(x)=-5 x $$

Step-by-Step Solution

Answer

The derivative of the function \( f(x) = -5x \) is \( f'(x) = -5 \).

Step 1: Write out the Definition of the Derivative

Start by writing out the limit definition of the derivative: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

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

Next, substitute \( f(x) \) and \( f(x+h) \) into the definition. We have the function \( f(x) = -5x \). So, \( f(x+h)= -5(x+h) = -5x -5h \), and \( f(x) = -5x \). Thus, the definition becomes: \( f'(x) = \lim_{h \to 0} \frac{-5x -5h - (-5x)}{h} \).

Step 3: Simplify the Expression

We simplify the expression to isolate the limit on one term. The expression simplifies to \( f'(x) = \lim_{h \to 0} \frac{-5h}{h} = \lim_{h \to 0} -5 \). This is a constant, so the limit as h approaches 0 is just -5.