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 $$ f^{\prime}(x)=u^{\prime}(x)+v^{\prime}(x) $$.

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UpStudy AI · Accepted Answer

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:
$$ y' = \frac{e^{x}}{5} - 7 + 4x $$
The derivative of the function is $$ y' = \frac{e^{x}}{5} - 7 + 4x $$.

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