### Finding the Derivative of the Given FunctionIn this example, we aim to determine the derivative of the function \( g(x) \).#### Function Definition\[ g(x) = \int_{3x}^{4x} \frac{u^2 - 1}{u^2 + 1} \, du \]#### Hint for Simplifying the Integral\[ \int_{a}^{b} f(u) \, du = \int_{a}^{0} f(u) \, du + \int_{0}^{b} f(u) \, du \]#### Computed Derivative\[ g'(x) = -3 \cdot \frac{9x^2 - 1}{9x^2 + 1} + 4 \cdot \frac{16x^2 - 5}{16x^2 + 5} \]### ExplanationTo find the derivative of the given integral, we can use the hint provided. Let's break down the steps:1. **Identify the function within the integral:** The integrand \( f(u) \) is \( \frac{u^2 - 1}{u^2 + 1} \).2. **Apply the hint:** The integral from \( 3x \) to \( 4x \) can be split using the hint provided.3. **Derivative of Integral Limits:** Apply the Fundamental Theorem of Calculus and the Leibniz rule to find the derivatives with respect to \( x \). - Differentiate the upper limit and lower limit terms separately.4. **Combine and simplify:** After differentiating the expressions individually, combine them to obtain \( g'(x) \).This step-by-step process helps in understanding how integrals can be handled using specific theorems and rules to find their derivatives effectively.

Name: ### Finding the Derivative of the Given FunctionIn this example, we a
Uploaded: 2024-03-20T06:47:22.000Z
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Description: ### Finding the Derivative of the Given FunctionIn this example, we aim to determine the derivative of the function \( g(x) \).#### Function Definition\[ g(x) = \int_{3x}^{4x} \frac{u^2 - 1}{u^2 + 1} \, du \]#### Hint for Simplifying the Integral\[