Question

Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).

• A. $-\frac{\sqrt{\pi}}{2} \cos(\frac{1}{\sqrt{2}})$
• B. $-\sqrt{\pi} \cos(\frac{1}{\sqrt{2}})$
• C. $-\frac{\sqrt{\pi}}{2} \sin(\frac{1}{\sqrt{2}})$
• D. $\sqrt{\frac{\pi}{2}} \sin(\frac{1}{\sqrt{2}})$

Hint:

When differentiating nested functions, apply the chain rule from the outside in. Differentiate the outermost function, keeping the inside intact, then multiply by the derivative of the next function inside, and so on, until you reach the innermost variable. Be careful with signs, especially when differentiating cosine.

Answer 1

The Correct Option is A

Step 1: Problem Identification:

The objective is to compute the derivative of a nested function, necessitating the iterative application of the chain rule. Subsequently, the derived expression will be evaluated at a given point.

Step 2: Governing Principle:

The chain rule dictates that for \( y = f(g(h(x))) \), the derivative is: \[ \frac{dy}{dx} = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \] The function is decomposed as \( y = \sin(u) \), \( u = \cos(v) \), and \( v = x^2 \).

Step 3: Derivation and Evaluation:

Given the function \( y = \sin(\cos(x^2)) \), we apply the chain rule for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{d}{dx} \sin(\cos(x^2)) \] \[ = \cos(\cos(x^2)) \cdot \frac{d}{dx}(\cos(x^2)) \] \[ = \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot \frac{d}{dx}(x^2) \] \[ = \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot (2x) \] \[ \frac{dy}{dx} = -2x \sin(x^2) \cos(\cos(x^2)) \] Evaluate this derivative at \( x = \frac{\sqrt{\pi}}{2} \). First, calculate the intermediate terms:

\[ x^2 = \left(\frac{\sqrt{\pi}}{2}\right)^2 = \frac{\pi}{4} \] Thus:

\[ \sin(x^2) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \] \[ \cos(x^2) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \] Substitute these values into the derivative formula:

\[ \frac{dy}{dx} \bigg|_{x=\frac{\sqrt{\pi}}{2}} = -2\left(\frac{\sqrt{\pi}}{2}\right) \cdot \sin\left(\frac{\pi}{4}\right) \cdot \cos\left(\cos\left(\frac{\pi}{4}\right)\right) \] \[ = -\sqrt{\pi} \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot \cos\left(\frac{1}{\sqrt{2}}\right) \] \[ = -\frac{\sqrt{\pi}}{\sqrt{2}} \cos\left(\frac{1}{\sqrt{2}}\right) \] This result is equivalent to \( -\sqrt{\frac{\pi}{2}} \cos\left(\frac{1}{\sqrt{2}}\right) \).

Step 4: Conclusive Result:

The derivative of the function evaluated at \( x = \frac{\sqrt{\pi}}{2} \) is \( -\frac{\sqrt{\pi}}{\sqrt{2}} \cos\left(\frac{1}{\sqrt{2}}\right) \). This corresponds to option (A).