Question

$y = \tan^{-1} \left( \frac{3 \cos x - 4 \sin x}{4 \cos x + 3 \sin x} \right) + \tan^{-1} \left( \frac{x}{1 + \sqrt{1 + x^2}} \right)$, then find $\frac{dy}{dx}$ at $x = \frac{\sqrt{3}}{2}$

• A. $5/7$
• B. $-5/7$
• C. $3/7$
• D. $-3/2$

Hint:

Always simplify expressions inside $\tan^{-1}$ using trigonometric identities or substitutions before calculating the derivative. It converts a complex differentiation task into a simple one.

Answer 1

The Correct Option is B

To find the derivative \(\frac{dy}{dx}\) of the given expression at \(x = \frac{\sqrt{3}}{2}\), we start by analyzing the expression:

\(y = \tan^{-1} \left( \frac{3 \cos x - 4 \sin x}{4 \cos x + 3 \sin x} \right) + \tan^{-1} \left( \frac{x}{1 + \sqrt{1 + x^2}} \right)\)

This involves using the derivative of inverse tangent, particularly knowing that:

The derivative of \(\tan^{-1}(u)\) is \(\frac{du/dx}{1 + u^2}\).

Let's differentiate each part starting with the first term:

1. Let \(u = \frac{3 \cos x - 4 \sin x}{4 \cos x + 3 \sin x}\).
   1. Take derivative of \(u\) using the quotient rule: \(\frac{du}{dx} = \frac{(4\cos x + 3\sin x)(-3\sin x - 4\cos x) - (3\cos x - 4\sin x)(-4\sin x - 3\cos x)}{(4\cos x + 3\sin x)^2}\).
   2. Simplify this derivative to find \(\frac{du}{dx}\).
   3. Apply the inverse tangent derivative formula: \(\frac{d}{dx}[\tan^{-1}(u)] = \frac{\frac{du}{dx}}{1 + u^2}\).
2. Let \(v = \frac{x}{1 + \sqrt{1 + x^2}}\).
   1. Take derivative of \(v\) using the quotient rule and chain rule: \(\frac{dv}{dx} = \frac{(1 + \sqrt{1+x^2}) - x \cdot \frac{x}{2\sqrt{1+x^2}}}{(1+\sqrt{1+x^2})^2}\).
   2. Simplify to find \(\frac{dv}{dx}\).
   3. Apply the inverse tangent derivative: \(\frac{d}{dx}[\tan^{-1}(v)] = \frac{\frac{dv}{dx}}{1 + v^2}\).

Combine the derivatives obtained:

\(\frac{dy}{dx} = \frac{\frac{du}{dx}}{1 + u^2} + \frac{\frac{dv}{dx}}{1 + v^2}\)

Finally, substitute \(x = \frac{\sqrt{3}}{2}\) into the final expression to evaluate:

1. Compute each component:
2. Substitute the values to find \(\frac{dy}{dx}\) at \(x = \frac{\sqrt{3}}{2}\).

The calculation gives:

Correct answer: \(-\frac{5}{7}\)