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.
Answer
The derivative is $\frac{3}{1 + 9x^2}$.
Answer for screen readers
The derivative of the function $y = \arctan(3x)$ is
$$ \frac{dy}{dx} = \frac{3}{1 + 9x^2} $$
Steps to Solve
- 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 $x$.
- Define $u$ and find its derivative
Here, let $u = 3x$. We need to find $\frac{du}{dx}$:
$$ \frac{du}{dx} = 3 $$
- Apply the chain rule
Now we substitute $u = 3x$ into the derivative formula. The derivative becomes:
$$ \frac{dy}{dx} = \frac{1}{1 + (3x)^2} \cdot \frac{du}{dx} $$
- 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
$$ \frac{dy}{dx} = \frac{3}{1 + 9x^2} $$
More Information
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 $x$.
Tips
- 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.
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