Question

If $y=\sin x\sin 2x$ and $t=\cos x$, then $\frac{dy}{dt}$ is:

• A. \(2(3t^2-1)\)
• B. \(1-3t^2\)
• C. \(\frac{1}{2}(1-3t^2)\)
• D. \((3t^2-1)\)
• E. \(2(1-3t^2)\)

Hint:

When one variable is given in terms of another, first rewrite the whole expression completely in that variable. Then differentiation becomes much easier.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves finding the derivative of a function \( y \) with respect to a variable \( t \), where both \( y \) and \( t \) are given as functions of another variable, \( x \). This is a case of parametric differentiation. We can use the chain rule to find \( \frac{dy}{dt} \).
Step 2: Key Formula or Approach:
The chain rule for parametric derivatives states:
\[ \frac{dy}{dt} = \frac{dy/dx}{dt/dx} \] We will find \( \frac{dy}{dx} \) and \( \frac{dt}{dx} \) separately and then take their ratio. Finally, we will express the result in terms of \( t \).
Step 3: Detailed Explanation:
First, find \( \frac{dt}{dx} \):
\[ t = \cos x \implies \frac{dt}{dx} = -\sin x \] Next, find \( \frac{dy}{dx} \). We have \( y = \sin x \sin 2x \). Let's use the product rule \( (uv)' = u'v + uv' \):
\[ \frac{dy}{dx} = (\cos x)(\sin 2x) + (\sin x)(\cos 2x \cdot 2) \] \[ \frac{dy}{dx} = \cos x \sin 2x + 2\sin x \cos 2x \] Now, use trigonometric identities to simplify this expression. We know \( \sin 2x = 2\sin x \cos x \) and \( \cos 2x = 2\cos^2 x - 1 \).
\[ \frac{dy}{dx} = \cos x (2\sin x \cos x) + 2\sin x (2\cos^2 x - 1) \] \[ \frac{dy}{dx} = 2\sin x \cos^2 x + 4\sin x \cos^2 x - 2\sin x \] \[ \frac{dy}{dx} = 6\sin x \cos^2 x - 2\sin x \] Factor out \( 2\sin x \):
\[ \frac{dy}{dx} = 2\sin x (3\cos^2 x - 1) \] Now, compute \( \frac{dy}{dt} \):
\[ \frac{dy}{dt} = \frac{dy/dx}{dt/dx} = \frac{2\sin x (3\cos^2 x - 1)}{-\sin x} \] Assuming \( \sin x \neq 0 \), we can cancel the terms:
\[ \frac{dy}{dt} = -2(3\cos^2 x - 1) \] Finally, substitute \( t = \cos x \) into the expression:
\[ \frac{dy}{dt} = -2(3t^2 - 1) = 2(1 - 3t^2) \] Step 4: Final Answer:
The derivative \( \frac{dy}{dt} \) is \( 2(1 - 3t^2) \). This corresponds to option (E).