What we need to do is take derivatives, and remember the general equation.

\(\displaystyle z=e^{x^2+4y^2}\sin(y)\)

\(\displaystyle w=g(x,y,z)\)

\(\displaystyle dw=g_xdx+g_ydy+g_zdz\)

When taking the derivative with respect to y recall that the product rule needs to be used.

\(\displaystyle dz=2xe^{x^2+4y^2}\sin(y)dx+(8y\sin(y)e^{x^2+4y^2}+cos(y)e^{x^2+4y^2}) dy\)

The total differential is defined as

\(\displaystyle dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy\)

We first find 

 \(\displaystyle \frac{\partial z}{\partial x}\)

by taking the derivative with respect to \(\displaystyle x\) and treating \(\displaystyle y\) as a constant.

 \(\displaystyle \frac{\partial z}{\partial x}=e^x\)

We then find 

 \(\displaystyle \frac{\partial z}{\partial y}\)

 by taking the derivative with respect to \(\displaystyle y\) and treating \(\displaystyle x\) as a constant.

 \(\displaystyle \frac{\partial z}{\partial y}=-sin(y)\)

We then substitute these partial derivatives into the first equation to get the total differential 

\(\displaystyle dz=e^x dx-sin(y)dy\)

The total differential is defined as

\(\displaystyle dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy\)

We first find \(\displaystyle \frac{\partial z}{\partial x}\) by taking the derivative with respect to \(\displaystyle x\) and treating \(\displaystyle y\) as a constant.

\(\displaystyle \frac{\partial z}{\partial x}=\frac{\partial }{\partial x}[e^{x^2+y^2}+sec(2x)]=2xe^{x^2+y^2}+2sec(2x)tan(2x)\)

We then find \(\displaystyle \frac{\partial z}{\partial y}\) by taking the derivative with respect to \(\displaystyle y\) and treating \(\displaystyle x\) as a constant.

\(\displaystyle \frac{\partial z}{\partial y}=\frac{\partial }{\partial y}[e^{x^2+y^2}+sec(2x)]=2ye^{x^2+y^2}\)

We then substitute these partial derivatives into the first equation to get the total differential 

\(\displaystyle dz=[2xe^{x^2+y^2}+2sec(2x)tan(2x)]dx+2ye^{x^2+y^2}dy\)

The total differential is defined as

\(\displaystyle dw=\frac{\partial w}{\partial x}dx+\frac{\partial w}{\partial y}dy+\frac{\partial w}{\partial z}dz\)

We first find \(\displaystyle \frac{\partial w}{\partial x}\) by taking the derivative with respect to \(\displaystyle x\) and treating the other variables as constants.

\(\displaystyle \frac{\partial w}{\partial x}=\frac{\partial }{\partial x}[e^{x+y}tan(4z)+sin(z)]=e^{x+y}tan(4z)\)

We then find \(\displaystyle \frac{\partial w}{\partial y}\) by taking the derivative with respect to \(\displaystyle y\) and treating the other variables as constants.

\(\displaystyle \frac{\partial w}{\partial y}=\frac{\partial }{\partial y}[e^{x+y}tan(4z)+sin(z)]=e^{x+y}tan(4z)\)

We then find \(\displaystyle \frac{\partial w}{\partial z}\) by taking the derivative with respect to \(\displaystyle z\) and treating the other variables as constants.

\(\displaystyle \frac{\partial w}{\partial z}=\frac{\partial }{\partial y}[e^{x+y}tan(4z)+sin(z)]=4e^{x+y}sec^2(4z)+cos(z)\)

We then substitute these partial derivatives into the first equation to get the total differential 

\(\displaystyle dw=e^{x+y}tan(4z)dx+e^{x+y}tan(4z)}dy +[4e^{x+y}sec^2(4z)+cos(z)]dz\)

The total derivative of a function of two variables is given by the following:

\(\displaystyle f_x dx+f_ydy\)

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

The partial derivatives of the function are

\(\displaystyle f_x=\sec(y)\)

\(\displaystyle f_y=x\sec(y)\tan(y)+24y^3\)

The derivatives were found using the following rules:

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

So, our final answer is

\(\displaystyle \sec(y)dx+(x\sec(y)\tan(y)+24y^3)dy\)

The total derivative of a function of two variables is given by

\(\displaystyle f_xdx+f_ydy\)

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

The partial derivatives are

\(\displaystyle f_x=\frac{1}{2\sqrt{x+y^2}}+\frac{2}{x}\)

\(\displaystyle f_y=\frac{y}{\sqrt{x+y^2}}\)

The derivatives were found using the following rules:

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

The total derivative of a function \(\displaystyle f(x, y, z)\) is given by

\(\displaystyle f_xdx+f_ydy+f_zdz\)

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

The partials are

\(\displaystyle f_x=-y^2\sin(xy^2)\)

\(\displaystyle f_y=-2xy\sin(xy^2)\)

\(\displaystyle f_z=2ze^{z^2}\)

The derivatives were found using the following rules:

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

The total derivative of a function is given by

\(\displaystyle f_xdx+f_ydy+f_zdz\)

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

The partial derivatives are

\(\displaystyle f_x=2xy\)

\(\displaystyle f_y=x^2\)

\(\displaystyle f_z=-\csc(z)\cot(z)\)

The derivatives were found using the following rules:

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

The differential of a function is given by

\(\displaystyle f_xdx+f_ydy+f_zdz\)

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

The partial derivatives are

\(\displaystyle f_x=5x^4\)

\(\displaystyle f_y=2yz\sec^2(y^2z)\)

\(\displaystyle f_z=y^2\sec^2(y^2z)\)

The derivatives were found using the following rules:

\(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}\), \(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)\), \(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)\), \(\displaystyle \frac{\mathrm{d} }{\mathrm{d} x} ax=a\)

The differential of a function is given by

\(\displaystyle f_xdx+f_ydy+f_zdz\)

The partial derivatives of the function are

\(\displaystyle f_x=\frac{2}{x}\)

\(\displaystyle f_y=ze^y\)

\(\displaystyle f_z=e^y\)

The derivatives were found using the following rules:

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