### Finding the Derivative of the FunctionGiven the function:\[ f(x) = \frac{1}{\sqrt[5]{(5x + 1)^2}} \]To find the derivative of the function \( f(x) \), follow these steps:1. **Rewrite the Function:** We start by expressing the given function in a more suitable form for differentiation. The original function is: \[ f(x) = \frac{1}{(5x + 1)^{2/5}} \] This can be rewritten using negative exponents: \[ f(x) = (5x + 1)^{-2/5} \]2. **Differentiate the Function:** Apply the chain rule to differentiate \( f(x) \). The chain rule states that if a function \( u(x) \) is differentiable and can be expressed as \( [u(x)]^n \), then: \[ \frac{d}{dx} [u(x)]^n = n [u(x)]^{n-1} \cdot \frac{du}{dx} \] Here, \( u(x) = 5x + 1 \) and \( n = -2/5 \). First, let’s find \( \frac{d}{dx}(5x + 1) \): \[ \frac{d}{dx}(5x + 1) = 5 \] Now, apply the chain rule: \[ f'(x) = \frac{d}{dx} (5x + 1)^{-2/5} = -\frac{2}{5} (5x + 1)^{-2/5 - 1} \cdot 5 \]3. **Simplify the Derivative:** Simplify the expression obtained: \[ f'(x) = -\frac{2}{5} (5x + 1)^{-7/5} \cdot 5 \] \[ f'(x) = -2 (5x + 1)^{-7/5} \]Therefore, the derivative of the given function is:\[ \boxed{ f'(x) = -2 (5x + 1)^{-7/5} } \]

Name: ### Finding the Derivative of the FunctionGiven the function:\[ f(x)
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: ### Finding the Derivative of the FunctionGiven the function:\[ f(x) = \frac{1}{\sqrt[5]{(5x + 1)^2}} \]To find the derivative of the function \( f(x) \), follow these steps:1. **Rewrite the Function:** We start by expressing the given function in