you might need: calculator\n$y = \\arctan(-4x)$\nevaluate $\\frac{dy}{dx}$ at $x = 3$.\nuse an exact…

you might need: calculator\n$y = \\arctan(-4x)$\nevaluate $\\frac{dy}{dx}$ at $x = 3$.\nuse an exact expression.

you might need: calculator\n$y = \\arctan(-4x)$\nevaluate $\\frac{dy}{dx}$ at $x = 3$.\nuse an exact expression.

Answer

Explanation:

Step1: Differentiate ( y = \arctan(-4x) ) using the chain rule

The derivative of ( \arctan(u) ) with respect to ( x ) is ( \frac{1}{1 + u^{2}}\cdot\frac{du}{dx} ). Let ( u=-4x ), then ( \frac{du}{dx}=-4 ). So ( \frac{dy}{dx}=\frac{-4}{1 + (-4x)^{2}}=\frac{-4}{1 + 16x^{2}} ).

Step2: Substitute ( x = 3 ) into the derivative

When ( x = 3 ), we have ( \frac{dy}{dx}\big|_{x = 3}=\frac{-4}{1+16\times3^{2}}=\frac{-4}{1 + 144}=\frac{-4}{145} ).

Answer:

( -\frac{4}{145} )