Question

If $y = \tan^{-1} \left[ \frac{12x - 64x^3}{1 - 48x^2} \right]$, then $dy/dx = \dots$

• A. $3/(1 + 16x^2)$
• B. $4/(1 + 16x^2)$
• C. $12/(1 + 16x^2)$
• D. $1/(1 + 16x^2)$

Hint:

Whenever you see a $1$ minus "something squared" in the denominator inside an inverse tangent, immediately look for the $\tan(2\theta)$ or $\tan(3\theta)$ substitution patterns. It turns a nightmare chain-rule problem into a 10-second derivative.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We must find the derivative of an inverse trigonometric function that has a complex rational polynomial argument. This requires trigonometric substitution to simplify it before differentiation.

Step 2: Key Formula or Approach:
The structure of the argument $\frac{A - B^3}{1 - 3B^2}$ strongly resembles the triple angle identity for tangent:
$$\tan(3\theta) = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$$
We need to find a substitution $4x = \tan \theta$ (or similar) that makes the argument match the formula exactly.

Step 3: Detailed Explanation:
Let's analyze the terms in the argument: $\frac{12x - 64x^3}{1 - 48x^2}$.
Notice that $64x^3 = (4x)^3$ and $48x^2 = 3(16x^2) = 3(4x)^2$.
Also, $12x = 3(4x)$.
Let's make the substitution: $\tan \theta = 4x \implies \theta = \tan^{-1}(4x)$.
Substitute this into the original function:
$$y = \tan^{-1} \left[ \frac{3(4x) - (4x)^3}{1 - 3(4x)^2} \right]$$
$$y = \tan^{-1} \left[ \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} \right]$$
The expression inside the brackets is exactly the expansion of $\tan(3\theta)$.
$$y = \tan^{-1} [\tan(3\theta)]$$
$$y = 3\theta$$
Replace $\theta$ with its $x$-equivalent:
$$y = 3\tan^{-1}(4x)$$
Now, differentiate with respect to $x$ using the chain rule ($\frac{d}{du}\tan^{-1}(u) = \frac{1}{1+u^2} \cdot \frac{du}{dx}$):
$$\frac{dy}{dx} = 3 \cdot \frac{1}{1 + (4x)^2} \cdot \frac{d}{dx}(4x)$$
$$\frac{dy}{dx} = 3 \cdot \frac{1}{1 + 16x^2} \cdot 4$$
$$\frac{dy}{dx} = \frac{12}{1 + 16x^2}$$

Step 4: Final Answer:
The derivative is $12/(1 + 16x^2)$, matching option (c).