Question

how do you solve this problem?

arctan (3x + 2)

i found the derivative using the product rule for (3x + 2) which is 3

now i get confused where i plug that in? i know next its chain rule but would i plug in the 3, (3x + 2), or both and where would they go respectively?

Explanation:

Step1: Recall the derivative formula for arctan(u)

The derivative of \(y = \arctan(u)\) with respect to \(x\) is \(\frac{dy}{dx}=\frac{1}{1 + u^{2}}\cdot\frac{du}{dx}\) (by the chain - rule). Here \(u = 3x+2\).

Step2: Find \(\frac{du}{dx}\)

If \(u = 3x + 2\), then \(\frac{du}{dx}=3\) (using the power rule: \(\frac{d}{dx}(ax + b)=a\) where \(a = 3\) and \(b = 2\)).

Step3: Substitute \(u\) and \(\frac{du}{dx}\) into the chain - rule formula

Substitute \(u = 3x+2\) and \(\frac{du}{dx}=3\) into \(\frac{dy}{dx}=\frac{1}{1 + u^{2}}\cdot\frac{du}{dx}\). We get \(\frac{dy}{dx}=\frac{3}{1+(3x + 2)^{2}}\).

Answer:

The derivative of \(y=\arctan(3x + 2)\) is \(\frac{3}{1+(3x + 2)^{2}}\)