differentiate with respect to x tan inverse[3tanx]

We are tasked with differentiating \(\tan^{-1}(3\tan x)\) with respect to \(x\).

Let \(y = \tan^{-1}(3\tan x)\).

### Step 1: Use Chain Rule
The chain rule tells us:
\[
\frac{dy}{dx} = \frac{d}{d(\theta)}\left( \tan^{-1}(\theta) \right) \cdot \frac{d}{dx}\left( 3\tan x \right)
\]
where \(\theta = 3\tan x\).

### Step 2: Differentiate the outer function
We know that the derivative of \(\tan^{-1}(u)\) with respect to \(u\) is:
\[
\frac{d}{du} \tan^{-1}(u) = \frac{1}{1 + u^2}
\]
So,
\[
\frac{d}{dx} \tan^{-1}(3\tan x) = \frac{1}{1 + (3\tan x)^2} \cdot \frac{d}{dx}(3\tan x)
\]

### Step 3: Differentiate the inner function
The derivative of \(3\tan x\) is:
\[
\frac{d}{dx}(3\tan x) = 3 \sec^2 x
\]

### Step 4: Combine the results
Now, combine the derivatives:
\[
\frac{dy}{dx} = \frac{1}{1 + 9\tan^2 x} \cdot 3 \sec^2 x
\]

### Final Answer:
\[
\frac{d}{dx} \left( \tan^{-1}(3\tan x) \right) = \frac{3 \sec^2 x}{1 + 9\tan^2 x}
\]

Would you like more details or have any questions about this?

Here are some related questions for further exploration:
1. What is the derivative of \(\tan^{-1}(x)\) in general?
2. How do you differentiate inverse trigonometric functions?
3. What is the derivative of \(\tan^{-1}(\sin x)\)?
4. What is the derivative of \(\sec^{-1}(3x)\)?
5. How does the chain rule work with trigonometric functions?

**Tip:** Always remember to apply the chain rule when you have a composition of functions!

Learn how to differentiate tan^(-1)(3tan(x)) with respect to x using the chain rule and derivative formulas for inverse trigonometric functions. The solution covers key differentiation concepts, including the application of the chain rule and derivatives of tan and tan^(-1). Suitable for high school students in Grades 11-12.

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