consider the function $$f(x) = 2x + 4y^2$$ and the curve $$\gamma(t) = (\sin t, \cos t), \quad t\in (0, 2\pi).$$ Then, what is the derivate of the composite function $f(\gamma(t))$ in the point $t_0 = \pi$ ?
2026-04-06 20:28:23.1775507303
trigonometric entity
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You can find this derivative either by:
direct substitution of $\gamma(t)$ into $f(x)$ and then evaluating $$\left.\frac{\mbox{d}}{\mbox{d}t}f\left(\gamma(t)\right)\right|_{t=\pi}$$
using the chain rule first: $$\frac{\mbox{d}f}{\mbox{d}x}\frac{\mbox{d}x}{\mbox{d}t}+\frac{\mbox{d}f}{\mbox{d}y}\frac{\mbox{d}y}{\mbox{d}t} = \ldots$$