Question

find y
y = \frac{3x - 2}{2x + 3}
y = \square

Explanation:

Step1: Find the first - derivative using the quotient rule

The quotient rule is \((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Let \(u = 3x - 2\), then \(u^\prime=3\); let \(v = 2x + 3\), then \(v^\prime = 2\).

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Step2: Find the second - derivative using the chain rule

The chain rule is \((f(g(x)))^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Let \(u = 2x+3\), \(y^\prime = 13u^{-2}\), then \(\frac{dy^\prime}{du}=-26u^{-3}\) and \(\frac{du}{dx}=2\).

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Answer:

\(-\frac{52}{(2x + 3)^{3}}\)