Find the derivative of the function. \[ y=\frac{e^{x}}{5}-7 x+2 x^{2} \] \( y^{\prime}=5 x^{4} e^{x 5}-7+4 x \) Please try again. Apply the Constant Function Rule, the Powers of \( x \) Rule, and the Sum Rule. . Constant Function Rule: If \( f(x)=c \), where \( c \) is a constant, then \( f^{\prime}(x)^{\prime}=0 \). - Powers of \( x \) Rule: If \( f(x)=x^{n} \) then \( f^{\prime}(x)=n x^{n-1} \) - Sum Rule: If \( f(x)=u(x)+v(x) \) then \( f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x) \).

Apologies for the confusion in your initial attempt. Let's correctly differentiate the function step by step using the **Constant Function Rule**, the **Powers of \( x \) Rule**, and the **Sum Rule**. Given the function: \[ y = \frac{e^{x}}{5} - 7x + 2x^{2} \] Let's break it down term by term. ### 1. Differentiate \(\frac{e^{x}}{5}\): - **Constant Multiple Rule**: If \( f(x) = c \cdot g(x) \), then \( f'(x) = c \cdot g'(x) \). Here, \( c = \frac{1}{5} \) and \( g(x) = e^{x} \). - **Derivative of \( e^{x} \)**: \( \frac{d}{dx} e^{x} = e^{x} \). Applying the rule: \[ \frac{d}{dx} \left( \frac{e^{x}}{5} \right) = \frac{1}{5} \cdot e^{x} = \frac{e^{x}}{5} \] ### 2. Differentiate \(-7x\): - **Powers of \( x \) Rule**: If \( f(x) = x^{n} \), then \( f'(x) = n \cdot x^{n-1} \). Here, \( n = 1 \). \[ \frac{d}{dx} (-7x) = -7 \cdot \frac{d}{dx} x = -7 \cdot 1 = -7 \] ### 3. Differentiate \(2x^{2}\): - **Powers of \( x \) Rule**: \( n = 2 \). \[ \frac{d}{dx} (2x^{2}) = 2 \cdot \frac{d}{dx} x^{2} = 2 \cdot 2x = 4x \] ### 4. Apply the Sum Rule: - **Sum Rule**: If \( f(x) = u(x) + v(x) + w(x) \), then \( f'(x) = u'(x) + v'(x) + w'(x) \). Combining the derivatives of each term: \[ y' = \frac{e^{x}}{5} - 7 + 4x \] ### Final Answer: \[ y' = \frac{e^{x}}{5} - 7 + 4x \]