Question

Differentiate the function \(\sin^2x\) with respect to the function \(e^{\cos x}\).

Hint:

Use \(\dfrac{du}{dv}=\dfrac{du/dx}{dv/dx}\), or rewrite \(u\) directly in terms of \(t=\cos x\).

Answer 1

Step 1: Understanding the Concept:
\"Differentiate \(u\) w.r.t. \(v\)\" means find \(\dfrac{du}{dv}=\dfrac{du/dx}{dv/dx}\), computing each ordinary derivative w.r.t. \(x\) first.

Step 2: Differentiating u = sin^2(x):
Let \(u=\sin^2x\). \(\dfrac{du}{dx}=2\sin x\cos x=\sin2x\).

Step 3: Differentiating v = e^{cos x}:
Let \(v=e^{\cos x}\). \(\dfrac{dv}{dx}=e^{\cos x}\cdot(-\sin x)=-\sin x\,e^{\cos x}\).

Step 4: Dividing:
\(\dfrac{du}{dv}=\dfrac{2\sin x\cos x}{-\sin x\,e^{\cos x}}=\dfrac{-2\cos x}{e^{\cos x}}\) (cancel \(\sin x\), valid where \(\sin x\neq0\)).

Final Answer:
\(\dfrac{du}{dv}=\boxed{-2\cos x\,e^{-\cos x}}\).