Answer to Question #345472 in Calculus for Gab

Question

Suppose z is a function of x and y and tan(√x + y) = e^z^2. Determine ∂z/∂x and ∂z/∂y.

Answer 1

$$\tan(\sqrt{x}+y)=e^{z^2}$$

Differentiate both sides with respect to $x$

$$\dfrac{1}{\cos ^2(\sqrt{x}+y)}(\dfrac{1}{2\sqrt{x}})=2ze^{z^2}\dfrac{\partial z}{\partial x}$$

$$\dfrac{\partial z}{\partial x}=\dfrac{1}{4ze^{z^2}\sqrt{x}\cos ^2(\sqrt{x}+y)}$$

$$\tan(\sqrt{x}+y)=e^{z^2}$$

Differentiate both sides with respect to $y$

$$\dfrac{1}{\cos ^2(\sqrt{x}+y)}(1)=2ze^{z^2}\dfrac{\partial z}{\partial y}$$

$$\dfrac{\partial z}{\partial y}=\dfrac{1}{2ze^{z^2}\cos ^2(\sqrt{x}+y)}$$