**Problem Statement:**Use the **limit definition of the derivative** to find a formula for \( f'(x) \) if \( f(x) = (g(x))^2 \), assuming that \( g \) is differentiable for all \( x \). *(Note: Your answer will be in terms of \( g(x) \) and \( g'(x) \). Do NOT use the Chain Rule in this case.)*---**Part (b):**Assume that \( f(x) = (g(x))^2 \), where \( g(2) = -3 \) and \( g'(2) = 4 \). Using your answer from (a), find the equation of the line tangent to \( y = f(x) \) at \( x = 2 \).--- *Explanation for Educational Use:*1. **Limit Definition of the Derivative:** This is a fundamental aspect of calculus used to determine the derivative of a function. It is defined as: \[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]2. **Given Function Analysis:** The function \( f(x) = (g(x))^2 \) suggests a function composition where \( f \) depends on another function \( g(x) \).3. **Finding the Derivative Using the Limit Definition:** - First, substitute \( f(x) \) into the limit definition: \[ f'(x) = \lim_{{h \to 0}} \frac{(g(x+h))^2 - (g(x))^2}{h} \] - Recognize the numerator as a difference of squares: \[ (g(x+h))^2 - (g(x))^2 = (g(x+h) - g(x))(g(x+h) + g(x)) \] - Substitute this back into the limit: \[ f'(x) = \lim_{{h \to 0}} \frac{(g(x+h) - g(x))(g(x+h) + g(x))}{h} \] - Apply the limit to \( g(x+h) - g(x) \), using the fact that \( g \) is differentiable: \[ f'(x) = g'(x

Name: **Problem Statement:**Use the **limit definition of the derivative**
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Problem Statement:**Use the **limit definition of the derivative** to find a formula for \( f'(x) \) if \( f(x) = (g(x))^2 \), assuming that \( g \) is differentiable for all \( x \). *(Note: Your answer will be in terms of \( g(x) \) and \( g'(x)