The chain rule for a composition

Method 1: direct composition.
$$f(\gamma(t)) = \cos^2 t \cdot \sin t.$$
Differentiating:
$$\frac{d}{dt}(\cos^2 t \sin t) = 2\cos t (-\sin t)\sin t + \cos^2 t \cos t = -2\cos t \sin^2 t + \cos^3 t.$$

Method 2: chain rule.
$$\nabla f(x,y) = (2xy, x^2), \qquad \gamma'(t) = (-\sin t, \cos t).$$
Hence
$$\frac{d}{dt}f(\gamma(t)) = 2\cos t \sin t \cdot (-\sin t) + \cos^2 t \cdot \cos t = -2\cos t \sin^2 t + \cos^3 t.$$

Conclusion. Both methods give the same result, illustrating the chain rule.