To find the derivative of the given function $y = \tan^{-1} \left( \frac{3x^2}{1 - 9x^2} \right)$, we'll proceed step by step.

Step 1: General formula

The derivative of $y = \tan^{-1}(u)$ is given by: $\frac{dy}{dx} = \frac{1}{1 + u^2} \cdot \frac{du}{dx},$ where $u$ is a function of $x$.

Here, $u = \frac{3x^2}{1 - 9x^2}$.

Step 2: Differentiate $u$

To differentiate $u = \frac{3x^2}{1 - 9x^2}$, we'll use the quotient rule: $\frac{du}{dx} = \frac{v \frac{d}{dx}(u) - u \frac{d}{dx}(v)}{v^2},$ where $u = 3x^2$ and $v = 1 - 9x^2$.

Compute derivatives:

$\frac{d}{dx}(3x^2) = 6x, \quad \frac{d}{dx}(1 - 9x^2) = -18x.$

Substitute these into the quotient rule: $\frac{du}{dx} = \frac{(1 - 9x^2)(6x) - (3x^2)(-18x)}{(1 - 9x^2)^2}.$

Simplify: $\frac{du}{dx} = \frac{6x - 54x^3 + 54x^3}{(1 - 9x^2)^2}.$

Combine terms: $\frac{du}{dx} = \frac{6x}{(1 - 9x^2)^2}.$

Step 3: Substitute back into the derivative formula

Now substitute $u = \frac{3x^2}{1 - 9x^2}$ and $\frac{du}{dx} = \frac{6x}{(1 - 9x^2)^2}$ into the formula: $\frac{dy}{dx} = \frac{1}{1 + \left(\frac{3x^2}{1 - 9x^2}\right)^2} \cdot \frac{6x}{(1 - 9x^2)^2}.$

Step 4: Simplify

The term $1 + \left(\frac{3x^2}{1 - 9x^2}\right)^2$ becomes: $1 + \frac{(3x^2)^2}{(1 - 9x^2)^2} = \frac{(1 - 9x^2)^2 + 9x^4}{(1 - 9x^2)^2}.$

Simplify the numerator: $(1 - 9x^2)^2 + 9x^4 = 1 - 18x^2 + 81x^4 + 9x^4 = 1 - 18x^2 + 90x^4.$

Thus: $1 + \left(\frac{3x^2}{1 - 9x^2}\right)^2 = \frac{1 - 18x^2 + 90x^4}{(1 - 9x^2)^2}.$

Finally, substitute everything into the derivative: $\frac{dy}{dx} = \frac{\frac{6x}{(1 - 9x^2)^2}}{\frac{1 - 18x^2 + 90x^4}{(1 - 9x^2)^2}}.$

Cancel out $(1 - 9x^2)^2$: $\frac{dy}{dx} = \frac{6x}{1 - 18x^2 + 90x^4}.$

Final Answer:

$\frac{dy}{dx} = \frac{6x}{1 - 18x^2 + 90x^4}.$

Let me know if you'd like further clarification or details! 😊

5 Related Questions:

1. How is the derivative of the arctangent function derived?
2. Can we find the critical points of the function $y = \tan^{-1} \left( \frac{3x^2}{1 - 9x^2} \right)$?
3. What happens to $\frac{dy}{dx}$ when $x$ approaches $\pm \frac{1}{3}$?
4. Can you explain how to expand $(1 - 9x^2)^2$ in more detail?
5. What are the applications of differentiating inverse trigonometric functions in real-life problems?

Tip:

Always check the domain of the given function, especially for rational and trigonometric expressions, as they can affect the existence of the derivative!