Calc I: Find the dervative: (4) \( y=(4 x+5)\left(\frac{3 x}{2 x^{2}+8}\right) \)
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To find the derivative of the function \( y = (4x + 5)\left(\frac{3x}{2x^{2} + 8}\right) \), we can use the product rule. The product rule states that if you have two functions \( u \) and \( v \), then the derivative \( y' = u'v + uv' \). Here, let \( u = (4x + 5) \) and \( v = \left(\frac{3x}{2x^{2} + 8}\right) \). Calculating \( u' \): \( u' = 4 \) Next, to find \( v' \), we will have to use the quotient rule for \( v = \frac{3x}{2x^2 + 8} \). According to the quotient rule: \( v' = \frac{(3)(2x^2 + 8) - (3x)(4x)}{(2x^2 + 8)^2} = \frac{6x^2 + 24 - 12x^2}{(2x^2 + 8)^2} = \frac{-6x^2 + 24}{(2x^2 + 8)^2} \) Now we plug \( u \), \( u' \), \( v \), and \( v' \) back into the product rule formula: \( y' = 4 \cdot \left(\frac{3x}{2x^{2} + 8}\right) + (4x + 5) \cdot \frac{-6x^2 + 24}{(2x^2 + 8)^2} \) Now, you can simplify and factor the expression as needed. Enjoy the winding journey of calculus!
