Question

Determine the first order partial derivative of the following functions:

1.z=in(x+t²).

2.f(x,y,z)=xy²e^-xz.

Answer 1

1.

$$z=\ln(x+t^2)$$

$$z_x=\dfrac{\partial z}{\partial x}=\dfrac{1}{x+t^2}$$

$$z_t=\dfrac{\partial z}{\partial t}=\dfrac{2t}{x+t^2}$$

2.

$$f(x,y,z)=xy^2e^{-xz}$$

$$f_x=\dfrac{\partial f}{\partial x}=y^2e^{-xz}-xy^2ze^{-xz}$$

$$f_y=\dfrac{\partial f}{\partial y}=2xye^{-xz}$$

$$f_z=\dfrac{\partial f}{\partial x}=-x^2y^2e^{-xz}$$