Question

The derivative of \(f(sinx)\) with respect to \(g(secx)\) at \(x = \frac{π}{4}\), given that \(f^'(\frac{1}{\sqrt{2}}) = 3\) and \(g^'(\sqrt{2}) = 1\) is.....

• A. \(\frac{\sqrt{3}}{2}\)
• B. \(\frac{3}{2}\)
• C. \(\frac{1}{3}\)
• D. \(\frac{-1}{3}\)

Hint:

Use the chain rule on both functions and divide: (df/dx)/(dg/dx).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To differentiate one function of \(x\) with respect to another, divide their derivatives with respect to \(x\): \(\frac{du}{dv} = \frac{du/dx}{dv/dx}\).

Step 2: Compute each:
\(\frac{d}{dx}f(\sin x) = f'(\sin x)\cos x\).
\(\frac{d}{dx}g(\sec x) = g'(\sec x)\sec x\tan x\).

Step 3: Evaluate at \(x = \frac\pi4\):
\(\sin\frac\pi4 = \frac1{\sqrt2}\), \(\cos\frac\pi4 = \frac1{\sqrt2}\), \(\sec\frac\pi4 = \sqrt2\), \(\tan\frac\pi4 = 1\).
Numerator: \(3 \times \frac{1}{\sqrt2}\). Denominator: \(1 \times \sqrt2 \times 1 = \sqrt2\).
\[ \frac{3/\sqrt2}{\sqrt2} = \frac{3}{2} \]

Step 4: Why the other options are wrong.
\(\frac{\sqrt3}{2}\) and \(\frac13\) do not follow from the given values. \(-\frac13\) has the wrong sign, since all factors here are positive.

Final Answer:
The derivative is \(\frac32\), option (B). \[ \boxed{\frac{3}{2}} \]