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: Write as a ratio of differentials:
Let $u = f(\sin x)$ and $v = g(\sec x)$. Then $\frac{du}{dv} = \frac{u'(x)}{v'(x)}$.

Step 2: Insert the data:
$u'(\frac\pi4) = 3 \cdot \frac{1}{\sqrt2}$ and $v'(\frac\pi4) = 1 \cdot \sqrt2 \cdot 1$.

Step 3: Divide:
$\frac{3}{\sqrt2 \cdot \sqrt2} = \frac32$.

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