Let $z = e ^ { x y }$ , where x = t cos t and y = t sin t.When t = 0, $\frac { d z } { d t }$ is

A) $\pi$
B) $- \pi$
C) $- 1$
D)0
E)1

Question

Let $z = e ^ { x y }$ , where x = t cos t and y = t sin t.When t = 0, $\frac { d z } { d t }$ is

A) $\pi$
B) $- \pi$
C) $- 1$
D)0
E)1

Quizplus · Accepted Answer

To find $\frac{dz}{dt}$ when $t = 0$, we use the chain rule. Given $z = e^{xy}$, $x = t \cos(t)$, and $y = t \sin(t)$, we first find $\frac{dz}{dx}$ and $\frac{dz}{dy}$, then multiply each by $\frac{dx}{dt}$ and $\frac{dy}{dt}$ respectively, and add them together. $\frac{dz}{dt} = \frac{dz}{dx} \cdot \frac{dx}{dt} + \frac{dz}{dy} \cdot \frac{dy}{dt}$$\frac{dz}{dx} = y e^{xy}$ and $\frac{dz}{dy} = x e^{xy}$$\frac{dx}{dt} = \cos(t) - t\sin(t)$ and $\frac{dy}{dt} = \sin(t) + t\cos(t)$At $t = 0$, $x = 0$, $y = 0$, $\frac{dx}{dt} = 1$, and $\frac{dy}{dt} = 0$.Thus, $\frac{dz}{dt} = 0 \cdot e^{0} \cdot 1 + 0 \cdot e^{0} \cdot 0 = 0$.

Let z=exyz = e ^ { x y }z=exy , Where X = T Cos T and Y =

Let $z = e ^ { x y }$ , Where X = T Cos T and Y =