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: Conceptualization:

The objective is to compute the derivative of a nested function using repeated application of the chain rule, followed by evaluation at a specific point.

Step 2: Methodology:

The chain rule for a triple composition \( y = f(g(h(x))) \) is given by: \[ \frac{dy}{dx} = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \] We are given \( y = \sin(u) \), \( u = \cos(v) \), and \( v = x^2 \).

Step 3: Derivation and Evaluation:

The composite function is \( y = \sin(\cos(x^2)) \). Applying the chain rule sequentially:

\[ \frac{dy}{dx} = \frac{d}{dx} \sin(\cos(x^2)) \] \[ = \cos(\cos(x^2)) \cdot \frac{d}{dx}(\cos(x^2)) \] \[ = \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot \frac{d}{dx}(x^2) \] \[ = \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot (2x) \] \[ \frac{dy}{dx} = -2x \sin(x^2) \cos(\cos(x^2)) \] Now, we evaluate this derivative at \( x = \frac{\sqrt{\pi}}{2} \). Calculate the components involving \( x \): \[ x^2 = \left(\frac{\sqrt{\pi}}{2}\right)^2 = \frac{\pi}{4} \] Subsequently:

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

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

Step 4: Conclusion:

The evaluated derivative at \( x = \frac{\sqrt{\pi}}{2} \) is \( -\frac{\sqrt{\pi}}{\sqrt{2}} \cos\left(\frac{1}{\sqrt{2}}\right) \). This aligns with option (A).