Question

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 incorrect answer question 4 of 26

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:

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