Step 1: Understanding the Concept:
We need to find the derivative of the sum of two functions, say \( f(x) + g(x) \), at \( x = 1 \).
Let \( f(x) = \cos^{-1}\left(\sin\sqrt{\frac{1+x}{2}}\right) \) and \( g(x) = x^x \).
We will differentiate them separately and sum their derivatives at \( x=1 \).
Step 2: Key Formula or Approach:
For \( f(x) \), use the trigonometric identity \( \sin \theta = \cos(\frac{\pi}{2} - \theta) \) to simplify the inverse cosine expression before differentiating.
For \( g(x) = x^x \), use logarithmic differentiation: let \( y = x^x \), then \( \ln y = x \ln x \).
Step 3: Detailed Explanation:
Let's simplify \( f(x) \):
\[ f(x) = \cos^{-1}\left(\sin\sqrt{\frac{1+x}{2}}\right) \]
Using \( \sin \theta = \cos(\frac{\pi}{2} - \theta) \):
\[ f(x) = \cos^{-1}\left(\cos\left(\frac{\pi}{2} - \sqrt{\frac{1+x}{2}}\right)\right) \]
For \( x = 1 \), the term \( \sqrt{\frac{1+1}{2}} = 1 \). The angle \( \frac{\pi}{2} - 1 \approx 0.57 \) radians, which is in the principal range \( [0, \pi] \) of \( \cos^{-1} \).
Thus, we can write:
\[ f(x) = \frac{\pi}{2} - \sqrt{\frac{1+x}{2}} \]
Now, differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = 0 - \frac{d}{dx}\left(\left(\frac{1+x}{2}\right)^{1/2}\right) \]
\[ f'(x) = - \frac{1}{2}\left(\frac{1+x}{2}\right)^{-1/2} \cdot \frac{d}{dx}\left(\frac{1+x}{2}\right) \]
\[ f'(x) = - \frac{1}{2\sqrt{\frac{1+x}{2}}} \cdot \frac{1}{2} \]
Evaluate at \( x = 1 \):
\[ f'(1) = - \frac{1}{2\sqrt{\frac{1+1}{2}}} \cdot \frac{1}{2} = - \frac{1}{2(1)} \cdot \frac{1}{2} = - \frac{1}{4} \]
Now for \( g(x) = x^x \):
Let \( y = x^x \implies \ln y = x \ln x \).
Differentiate implicitly:
\[ \frac{1}{y} \cdot \frac{dy}{dx} = 1 \cdot \ln x + x \cdot \frac{1}{x} \]
\[ \frac{dy}{dx} = y (\ln x + 1) = x^x (\ln x + 1) \]
So, \( g'(x) = x^x (\ln x + 1) \).
Evaluate at \( x = 1 \):
\[ g'(1) = 1^1 (\ln 1 + 1) = 1 \cdot (0 + 1) = 1 \]
Finally, sum the derivatives:
\[ \frac{d}{dx}[f(x) + g(x)] \Big|_{x=1} = f'(1) + g'(1) = - \frac{1}{4} + 1 = \frac{3}{4} \]
Step 4: Final Answer:
The value of the derivative is \( \frac{3}{4} \).