Question

If $y = \sin^{-1}\left[\cos \sqrt{\frac{1+x}{2}}\right] + x^x$, then $\frac{dy}{dx}$ at $x=1$ is

• A. $\frac{5}{4}$
• B. $-\frac{1}{4}$
• C. $\frac{3}{4}$
• D. $-\frac{5}{4}$

Hint:

Memorizing $\frac{d}{dx}(x^x) = x^x(1 + \log x)$ is highly recommended for competitive exams. It saves the entire process of taking logarithms on both sides and performing implicit differentiation.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We must find the derivative of a composite function at a specific point, $x=1$. The function is the sum of an inverse trigonometric term and an exponential-algebraic term.

Step 2: Key Formula or Approach:
We can simplify the first term using the co-function identity: $\cos \theta = \sin\left(\frac{\pi}{2} - \theta\right)$. For the second term, we use logarithmic differentiation or the standard result $\frac{d}{dx}(x^x) = x^x(1 + \log x)$.

Step 3: Detailed Explanation:
Let $y = y_1 + y_2$, where $y_1 = \sin^{-1}\left[\cos \sqrt{\frac{1+x}{2}}\right]$ and $y_2 = x^x$.
Simplify $y_1$:
$$y_1 = \sin^{-1}\left[\sin\left(\frac{\pi}{2} - \sqrt{\frac{1+x}{2}}\right)\right]$$ Since $\sqrt{\frac{1+x}{2}}$ at $x=1$ is $1$ radian, which falls well within the principal range, we can cancel the inverse sine and sine:
$$y_1 = \frac{\pi}{2} - \sqrt{\frac{1+x}{2}} = \frac{\pi}{2} - \frac{1}{\sqrt{2}}(1+x)^{1/2}$$ Differentiate $y_1$ with respect to $x$:
$$\frac{dy_1}{dx} = 0 - \frac{1}{\sqrt{2}} \cdot \frac{1}{2}(1+x)^{-1/2} = -\frac{1}{2\sqrt{2}\sqrt{1+x}}$$ Evaluate $\frac{dy_1}{dx}$ at $x = 1$:
$$\left. \frac{dy_1}{dx} \right|_{x=1} = -\frac{1}{2\sqrt{2}\sqrt{2}} = -\frac{1}{2(2)} = -\frac{1}{4}$$ Now for $y_2 = x^x$. Its standard derivative is:
$$\frac{dy_2}{dx} = x^x(1 + \log x)$$ Evaluate $\frac{dy_2}{dx}$ at $x = 1$:
$$\left. \frac{dy_2}{dx} \right|_{x=1} = 1^1(1 + \log 1) = 1(1 + 0) = 1$$ Combine both derivatives to get $\frac{dy}{dx}$:
$$\left. \frac{dy}{dx} \right|_{x=1} = \left. \frac{dy_1}{dx} \right|_{x=1} + \left. \frac{dy_2}{dx} \right|_{x=1}$$ $$\left. \frac{dy}{dx} \right|_{x=1} = -\frac{1}{4} + 1 = \frac{3}{4}$$

Step 4: Final Answer:
The value of the derivative at $x=1$ is $\frac{3}{4}$, matching option (C).