### Problem Statement**Objective:** Find the derivative \( f'(x) \) of the given function.**Function Definition:**\[ f(x) = \frac{e^x}{5x^2 + 2} \]### ExplanationTo solve for \( f'(x) \), you'll need to apply the quotient rule for differentiation, which is used when finding the derivative of a division of two functions. The formula for the quotient rule is:\[\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}\]Where:- \( u = e^x \) and \( u' = \frac{d}{dx}(e^x) = e^x \)- \( v = 5x^2 + 2 \) and \( v' = \frac{d}{dx}(5x^2 + 2) = 10x \)### Solution Steps1. **Apply the Quotient Rule:**\[f'(x) = \frac{e^x(5x^2 + 2) - e^x \cdot 10x}{(5x^2 + 2)^2}\]2. **Simplify the Expression:**Now, calculate the expression in the numerator:\[f'(x) = \frac{e^x(5x^2 + 2 - 10x)}{(5x^2 + 2)^2}\]3. **Further Simplification:**The result can be further simplified within the brackets:\[5x^2 + 2 - 10x = 5x^2 - 10x + 2\]Thus, the derivative \( f'(x) \) becomes:\[f'(x) = \frac{e^x (5x^2 - 10x + 2)}{(5x^2 + 2)^2}\]The derivative \( f'(x) \) gives the rate of change of the function \( f(x) \) with respect to \( x \), using quotient rule differentiation.### ConclusionThis problem effectively utilizes the quotient rule of differentiation, an important concept in calculus for calculating derivatives of functions involving division.

Name: ### Problem Statement**Objective:** Find the derivative \( f'(x) \) o
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: ### Problem Statement**Objective:** Find the derivative \( f'(x) \) of the given function.**Function Definition:**\[ f(x) = \frac{e^x}{5x^2 + 2} \]### ExplanationTo solve for \( f'(x) \), you'll need to apply the quotient rule for differentiation, w