To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must take the derivative of the function with respect to y:

$\displaystyle f_y=2xyz+x\sec(y)\tan(y)$

The derivative was found using the following rules:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Finally, we take the derivative of the above function with respect to x:

$\displaystyle f_{yx}=2yz+\sec(y)\tan(y)$

The derivative was found using rules above as well as

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} \sec(x)=\sec(x)\tan(x)$

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the derivative of the function with respect to x:

$\displaystyle f_y=zxe^{xy}$

The derivative was found using the following rule:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u \frac{\mathrm{d} u}{\mathrm{d} x}$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Finally, we find the derivative of the above function with respect to x:

$\displaystyle f_{yx}=ze^{xy}+zx^2e^{xy}$

We used the above rules to find the derivative as well as the following:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must first find the derivative of the function with respect to y:

$\displaystyle f_y=x\cos(z^2)+z^3$

The derivative was found using the following rules:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Finally, we take the derivative of the above function with respect to z:

$\displaystyle f_{yz}=-2xz\sin(z^2)+3z^2$

The derivative was found using the above rule as well as the following:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

We must find the derivative of the function with respect to y:

$\displaystyle f_y=z^2+2xye^{y^2}$

The derivative was found using the following rules:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Using the formula for a plane $\displaystyle A(x-x_0)+B(y-y_0)+C(z-z_0)=0$, where the point given is $\displaystyle (x_0, y_0, z_0)$ and the normal vector is $\displaystyle \left \langle A,B,C\right \rangle$. Plugging in the known values, you get $\displaystyle 5(x-0)+2(y-2)+1(z-0)=0$. Manipulating this equation through algebra gives you the answer $\displaystyle 5x+2y+z=4$.

In order to solve, you must take a total of three derivatives: the first is $\displaystyle \frac{\partial }{\partial y}$, then again $\displaystyle {\frac{\partial }{\partial y}}$, and finally $\displaystyle \frac{\partial }{\partial z}$,in that order (the notation in the problem statement dictates that). The first derivative you obtain will be $\displaystyle 6y^5z+4y^3x^2$. The subsequent derivative is $\displaystyle 30y^4z+12y^2x^2$. The final derivative with respect to z is $\displaystyle 30y^4$. The rule used for all derivatives is $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}x^n= nx^{n-1}$, and we treat all other variables as constants.

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

So, we must find the partial derivative with respect to z:

$\displaystyle f_z=-2z\csc(z^2)\cot(z^2)+xy$

The derivative was found using the following rules:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} \csc(x)=-\csc(x)\cot(x)$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$,

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative of the function with respect to x:

$\displaystyle f_x=4x^3yz+\frac{\cos(z)}{x}$

The following rules were used:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} \ln(x)=\frac{1}{x}$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Next, we take the partial derivative with respect to x of the above function:

$\displaystyle f_{xx}=12x^2yz-\frac{\cos(z)}{x^2}$

The rules used are stated above.

Finally, we take the partial derivative of the function above with respect to z:

$\displaystyle f_{xxz}=12x^2y+\frac{\sin(z)}{x^2}$

The rules we used are stated above, as well as

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x}\cos(x)=-\sin(x)$

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivative of the function with respect to x is

$\displaystyle f_x=5\cos(z)+e^{xy}+xye^{xy}$

and was found using the following rules:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$, $\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$,$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$, 

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we must find the partial derivative of the function with respect to x:

$\displaystyle f_x=6z^2\cos(y)$

The derivative was found using the following rule:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Next, we find the partial derivative of the above function with respect to z:

$\displaystyle f_{xz}=12z\cos(y)$

We used the above rule and the following rule:

$\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Finally, we find the partial derivative of the above function with respect to z:

$\displaystyle f_{xzz}=12\cos(y)$

The above two rules were used.