Question

Multiple Choice Which of the following is \(d y / d x\) if \(y=\cos ^{2}\left(x^{3}+x^{2}\right) ?\) (A) \(-2\left(3 x^{2}+2 x\right)\) (B) \(-\left(3 x^{2}+2 x\right) \cos \left(x^{3}+x^{2}\right) \sin \left(x^{3}+x^{2}\right)\) (C) \(-2\left(3 x^{2}+2 x\right) \cos \left(x^{3}+x^{2}\right) \sin \left(x^{3}+x^{2}\right)\) (D) 2\(\left(3 x^{2}+2 x\right) \cos \left(x^{3}+x^{2}\right) \sin \left(x^{3}+x^{2}\right)\) (E) 2\(\left(3 x^{2}+2 x\right)\)

Short answer

The correct answer is (C)

Step-by-step solution

Identify the outer and inner function

The given function is \(y=\cos ^{2}\left(x^{3}+x^{2}\right)\). Here, the outer function is \(\cos ^{2}(u)\) and the inner function is \(u=x^{3}+x^{2}\). We need to find the derivative of the composite function.

Differentiate the outer function

The derivative of \(\cos ^{2}(u)\) with respect to \(u\) is \(-2\cos(u)\sin(u)\) using the chain rule for \(\cos^2(u)\).

Differentiate the inner function

The derivative of \(x^{3}+x^{2} = 3x^{2}+2x\) using the power rule.

Apply the chain rule for differentiation

To differentiate the composite function \(y=\cos ^{2}\left(x^{3}+x^{2}\right)\), use the Chain Rule for differentiation, which states that the derivative of a composite function is the derivative of the outer function, multiplied by the derivative of the inner function. \[\frac{dy}{dx} = -2\cos(u)\sin(u) * \frac{du}{dx}\] Substitute \(u=x^{3}+x^{2}\) and \(\frac{du}{dx}=3x^{2}+2x\) into the equation.

Get the final answer

Substitute \(u\) and the derivative of \(u\) back into the equation, get: \(\frac{dy}{dx} = -2\cos\left(x^{3}+x^{2}\right)\sin\left(x^{3}+x^{2}\right) * \left(3x^{2}+2x\right)\)