if y = cos³(x²), then dy/dx = -6x cos²(x²) sin x² -6x cos²(x²) 6x cos²(x²) sin x² -3 cos²(x²) sin x…

Explanation:

Step1: Identify the outer - function and inner - function

Let (u = \cos(x^{2})), then (y = u^{3}).

Step2: Differentiate (y) with respect to (u)

Using the power rule (\frac{d}{du}(u^{n})=nu^{n - 1}), we have (\frac{dy}{du}=3u^{2}).

Step3: Differentiate (u) with respect to (x)

Let (t=x^{2}), then (u = \cos(t)). First, (\frac{dt}{dx}=2x), and (\frac{du}{dt}=-\sin(t)). By the chain - rule (\frac{du}{dx}=\frac{du}{dt}\cdot\frac{dt}{dx}=-\sin(x^{2})\cdot2x=- 2x\sin(x^{2})).

Step4: Use the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx})

Substitute (u = \cos(x^{2})), (\frac{dy}{du}=3u^{2}) and (\frac{du}{dx}=-2x\sin(x^{2})) into the chain - rule formula. We get (\frac{dy}{dx}=3\cos^{2}(x^{2})\cdot(-2x\sin(x^{2}))=-6x\cos^{2}(x^{2})\sin(x^{2})).

Answer:

(-6x\cos^{2}(x^{2})\sin(x^{2}))