Question:

Step 1: Simplify the innermost inverse trigonometric function. For $x \in [-\pi/2, \pi/2]$, we have the property: $$\sin^{-1}(\sin x) = x$$ Since $x = \pi/4$ falls within this interval, we can apply this property: $$\sin^{-1}(\sin x) = x$$ Step 2: Simplify the exponential and logarithm composition. We now evaluate $e^{\sin^{-1}\sin x}$: $$e^{\sin^{-1}\sin x} = e^x$$ Taking the natural logarithm of this expression: $$\ln(e^x) = x$$ Step 3: Simplify the logarithm base 2 expression. We need to evaluate $\log_2 2^{\ln e^{\sin^{-1}\sin x}}$. From Step 2, we found that $\ln e^{\sin^{-1}\sin x} = x$, so: $$\log_2 2^x = x$$ This uses the property that $\log_b b^a = a$. Step 4: Simplify the inverse cosine function. Now we evaluate the outermost function: $$y = \cos^{-1}(\cos(x))$$ For $x \in [0, \pi]$, the inverse cosine function satisfies: $$\cos^{-1}(\cos x) = x$$ Since $x = \pi/4 \in [0, \pi]$, we have: $$y = x$$ Step 5: Find the derivative and evaluate at $x = \pi/4$. Since $y = x$, we differentiate both sides with respect to $x$: $$\frac{dy}{dx} = 1$$ This derivative is constant for all values of $x$ in the valid domain, including at $x = \pi/4$. Therefore, the value of $\dfrac{dy}{dx}$ at $x = \dfrac{\pi}{4}$ is $\boxed{1}$. The correct answer is **Option 4**.