To find \(\displaystyle f_{xy}\) of the function, you must take two consecutive partial derivatives:

\(\displaystyle \frac{\partial }{\partial x}(e^x+x\cos(y))=e^x+\cos(y)\)

\(\displaystyle \frac{\partial }{\partial y}(e^x+\cos(y))=-\sin(y)\)

When calculating the partial derivative with respect to the variable \(\displaystyle x\) of a function \(\displaystyle f\) of more than one variable, apply the standard rules for differentiating a function \(\displaystyle f(x)\) of a single variable, and treat the other variables as constants. In this case, we have:

\(\displaystyle f(x,y)=x^2y^2+\frac{x}{x+y}\)

We are being asked to differentiate \(\displaystyle f(x,y)\) with respect to \(\displaystyle x\), so we treat the variable \(\displaystyle y\) as a constant and apply the sum and quotient rules of differentiation to find the partial derivative \(\displaystyle \frac{\partial f}{\partial x}\), as shown:

\(\displaystyle \frac{\partial f}{\partial x}=2x(y^2)+\frac{(1)(x+y)-(1)(x)}{(x+y)^2}\)

\(\displaystyle =2xy^2+\frac{x+y-x}{(x+y)^2}\)

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

Since \(\displaystyle y\) is treated as a constant, we can factor out \(\displaystyle y\) from the sum of these two terms and simplify the expression:

\(\displaystyle \frac{\partial f}{\partial x}=y(2xy+\frac{1}{(x+y)^2})\)

When calculating the partial derivative with respect to the variable \(\displaystyle x\) of a function \(\displaystyle f\) of more than one variable, apply the standard rules for differentiating a function \(\displaystyle f(x)\) of a single variable, and treat the other variables as constants. In this case, we are given \(\displaystyle f(x,y,z)=x^2y^4z^5\), and its partial derivative with respect to \(\displaystyle z\) can be calculated by treating \(\displaystyle x\) and \(\displaystyle y\) and constants and differentiating \(\displaystyle z^5\):

\(\displaystyle \frac{\partial f}{\partial z}=5x^2y^4z^4\).

Now substitute the values for the point \(\displaystyle (3,1,-2)\) into \(\displaystyle \frac{\partial f}{\partial z}\) to calculate its value at that point:

\(\displaystyle F_z(x,y,z)=5(3)^2(1)^4(-2)^4=5\times9\times1\times16=720\).

When calculating the partial derivative with respect to the variable \(\displaystyle y\) of a function \(\displaystyle f\) of more than one variable, apply the standard rules for differentiating a function \(\displaystyle f(y)\) of a single variable, and treat the other variables as constants. In this case, we have:

\(\displaystyle f(x,y,z)=7e^x^z\ln(y)\)

We are being asked to differentiate \(\displaystyle f(x,y,z)\) with respect to \(\displaystyle y\), so we treat the variables \(\displaystyle x\) and \(\displaystyle z\) as constants, recognize that the term \(\displaystyle 7e^x^z\) is now just a constant, and apply the rule of differentiation for the natural logarithm to find the partial derivative \(\displaystyle \frac{\partial f}{\partial y}\), as shown:

\(\displaystyle \frac{\partial f}{\partial y}=7e^x^z(\ln(y))'\)

\(\displaystyle =7e^x^z(\frac{1}{y})\)

\(\displaystyle =\frac{7e^x^z}{y}\)

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 find the partial derivative of the function with respect to x:

\(\displaystyle f_x=2xy^2e^{x^2y^2}+\tan(y)\)

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

\(\displaystyle f_{xy}=4xye^{x^2y^2}+4x^3y^3e^{x^2y^2}+\sec^2(y)\)

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 find the partial derivative of the function with respect to x:

\(\displaystyle f_x=-y^3\sin(xy)\)

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

\(\displaystyle f_{xx}=-y^4\cos(xy)\)

To find \(\displaystyle f_x\) of the function, you take the partial derivative of the function with respect to x

\(\displaystyle \frac{\partial }{\partial x}(xy+z\cos(x))=y-z\sin(x)\)

To find \(\displaystyle f_{yy}\) of the function, you must take two consecutive partial derivatives

\(\displaystyle \frac{\partial }{\partial y}(e^{2xy}+2xy^3)=2xe^{2xy}+6xy^2\)

\(\displaystyle \frac{\partial }{\partial y}(2xe^{2xy}+6xy^2)=4x^2e^{2xy}+12xy\)

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 take the partial derivative of the function with respect to x:

\(\displaystyle f_x=y^2+2xe^y\)

Next, we take the partial derivative of this function with respect to y:

\(\displaystyle f_{xy}=2y+2xe^y\)

Finally, we square this:

\(\displaystyle (f_{xy})^2=(2y+2xe^y)^2=4y^2+8xye^y+4x^2e^{2y}\)

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 take the partial derivative of the function with respect to x:

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

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

\(\displaystyle f_{xx}=\frac{2}{y^2}-\frac{y}{x^2}\)