Question

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$y = \arctan(-4x)$
evaluate $\frac{dy}{dx}$ at $x = 3$.
use an exact expression.

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:

\( -\frac{4}{145} \)