# What is the derivative of arctan(3x)?

#### Understand the Problem

The question is asking for the derivative of the function arctan(3x). To solve this, we will apply the chain rule in calculus, which helps us find the derivative of composite functions.

The derivative is $\frac{3}{1 + 9x^2}$.

The derivative of the function $y = \arctan(3x)$ is

$$\frac{dy}{dx} = \frac{3}{1 + 9x^2}$$

#### Steps to Solve

1. Identify the function and derivative formula

We are given the function $y = \arctan(3x)$. The derivative of $y = \arctan(u)$ with respect to $x$ is given by:

$$\frac{dy}{dx} = \frac{1}{1 + u^2} \cdot \frac{du}{dx}$$

where $u$ is a function of $x$.

1. Define $u$ and find its derivative

Here, let $u = 3x$. We need to find $\frac{du}{dx}$:

$$\frac{du}{dx} = 3$$

1. Apply the chain rule

Now we substitute $u = 3x$ into the derivative formula. The derivative becomes:

$$\frac{dy}{dx} = \frac{1}{1 + (3x)^2} \cdot \frac{du}{dx}$$

1. Simplify the expression

Now we simplify the expression where $u = 3x$:

$$\frac{dy}{dx} = \frac{1}{1 + 9x^2} \cdot 3$$

This gives us:

$$\frac{dy}{dx} = \frac{3}{1 + 9x^2}$$

The derivative of the function $y = \arctan(3x)$ is

$$\frac{dy}{dx} = \frac{3}{1 + 9x^2}$$

This result comes from applying the chain rule, which is particularly useful when dealing with compositions of functions like $y = \arctan(u)$ where $u$ is a function of $x$. The derivative tells us the slope of the function $y = \arctan(3x)$ at any point $x$.
• Forgetting to apply the chain rule: Many people forget that they need to differentiate the inner function ($u = 3x$) as well when using the chain rule.
• Confusing the derivative of $\arctan(u)$: The derivative must be remembered as $\frac{1}{1 + u^2}$, and it's easy to mix it up with other trigonometric derivatives.