Question

$[\frac{d}{dx}((sin~x)^{cos~x})]_{x=7/4}=$}

• A. $(\frac{1}{\sqrt{2}})^{(\frac{\sqrt{2}+1}{\sqrt{2}})}(1+log\sqrt{2})$
• B. $(\frac{1}{\sqrt{2}})^{\frac{1}{\sqrt{2}}}(1+log\sqrt{2})$
• C. $(\frac{1}{\sqrt{2}})^{\frac{1}{\sqrt{2}}}(1-log\sqrt{2})$
• D. $(\frac{1}{\sqrt{2}})^{(\frac{\sqrt{2}+1}{\sqrt{2}})}(1-log\sqrt{2})$ \bigskip

Hint:

Always convert variable powers using logarithmic differentiation.

Answer 1

The Correct Option is B

Concept: We apply logarithmic differentiation for variable power functions.

Step 1: Take logarithm on both sides.
\[ y=(sin~x)^{cos~x} \Rightarrow log~y = cos~x \cdot log(sin~x) \]

Step 2: Differentiate implicitly.
\[ \frac{1}{y}y' = -sin~x \cdot log(sin~x) + cos~x \cdot \frac{cos~x}{sin~x} \]

Step 3: Substitute $x=\frac{\pi}{4}$.
\[ sin~x = cos~x = \frac{1}{\sqrt{2}} \]

Step 4: Simplify final expression.
\[ y' = (\frac{1}{\sqrt{2}})^{\frac{1}{\sqrt{2}}}(1+log\sqrt{2}) \]