AP Calculus BC - Differentials

Find the total derivative of the following function:

$f(x, y,z)=x^2y+\csc(z)$

$2xydx+x^2dy-\csc(z)\cot(z)dz$

$2xydx+x^2dy+\csc(z)\cot(z)dz$

$2xdx-\csc(z)\cot(z)dz$

$2xy+x^2-\csc(z)\cot(z)$

Explanation

The total derivative of a function is given by

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

The partial derivatives are

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

The derivatives were found using the following rules:

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

If $z=f(x,y)=x^6+3xy-\sin(xy)$ , calculate the total differential .

$dz=(6x^5+3y-y\cos(xy))dx+(3x-x\cos(xy))dy$

$dz=(6x^5+3y-\cos(xy))dx+(3x-\cos(xy))dy$

$dz=(\frac{x^7}{7}+3y-y\cos(xy))dx+(\frac{3x^2}{2}-x\cos(xy))dy$

$dz=(6x^5+3y+y\cos(xy))dx+(3x+x\cos(xy))dy$

$dz=(6x^5-3y+y\cos(xy))dx+(3+x\cos(xy))dy$

Explanation

The total differential of a function is defined as the sum of the partial derivatives of with respect to each of its variables; that is,

$dz=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$

In this case, $z=f(x,y)=x^6+3xy-\sin(xy)$ , and so we use the sum rule, the rule for derivatives of a variable raised to a power, the rule for the derivative of $\sin(x)$ , and the chain rule to calculate the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ , as shown:

$\frac{\partial f}{\partial x}=6x^5+3y-y\cos(xy)$ ,

$\frac{\partial f}{\partial y}=3x-x\cos(xy)$ .

In both cases, we treated the variable not being differentiated as a constant, and applied the chain rule to $\sin(xy)$ to calculate its partial derivatives. Now that $\frac{\partial f}{\partial x}dx$ and $\frac{\partial f}{\partial y}dy$ have been calculated, all that remains is to substitute them into the definition of the total derivative:

$dz=(6x^5+3y-y\cos(xy))dx+(3x-x\cos(xy))dy$

Compute the differentials for the following function.

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

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

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

$dz=2xe^{x^2+4y^2}\sin(y)dx$

$dz=8y\cos(y)e^{x^2+4y^2} dy$

$dz=2xe^{x^2+4y^2}\sin(y)$

Explanation

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

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

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

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

Find the differential of the following function:

$f(x,y,z)=\cos^2(xy)+\sin(z)$

$-2y\cos(xy)\sin(xy)dx-2x\cos(xy)\sin(xy)dy+\cos(z)dz$

$2y\cos(xy)\sin(xy)dx+2x\cos(xy)\sin(xy)dy+\cos(z)dz$

$-y\cos(xy)\sin(xy)dx-x\cos(xy)\sin(xy)dy+\cos(z)dz$

$-2x\cos(xy)\sin(xy)dx-2y\cos(xy)\sin(xy)dy+\cos(z)dz$

$-2y\cos(xy)\sin(xy)dx-2x\cos(xy)\sin(xy)dy-\cos(z)dz$

Explanation

The differential of a function is given by

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

The partial derivatives are

$f_x=-2y\cos(xy)\sin(xy)$

$f_y=-2x\cos(xy)\sin(xy)$

$f_z=\cos(z)$

Find the differential of the following function:

Explanation

The differential of the function is given by

The partial derivatives are

, ,

Find the total differential , , of the function

Explanation

The total differential is defined as

$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$

We first find

$\frac{\partial z}{\partial x}$

by taking the derivative with respect to and treating as a constant.

$\frac{\partial z}{\partial x}=e^x$

We then find

$\frac{\partial z}{\partial y}$

by taking the derivative with respect to and treating as a constant.

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

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

Find the differential of the function:

Explanation

The differential of the function is given by

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

The partial derivatives are

Find the differential of the following function:

$f(x,y,z)=7xy+6x^2y^3e^{z^2}$

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

$(7x+12xy^2e^{z^2})dx+(7y+18x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+12x^2y^2e^{z^2})dy+(12x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(6x^2y^3ze^{z^2})dz$

$(7y+12xy^2e^{z^2})dx+(7x+18x^2y^2e^{z^2})dy+(12x^2y^3z^2e^{z^2})dz$

Explanation

The differential of a function is given by

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

The partial derivatives are

$f_x=7y+12xy^2e^{z^2}$

$f_y=7x+18x^2y^2e^{z^2}$

$f_z=12x^2y^3ze^{z^2}$

Compute the differentials for the following function.

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

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

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

$dz=2xe^{x^2+4y^2}\sin(y)dx$

$dz=8y\cos(y)e^{x^2+4y^2} dy$

$dz=2xe^{x^2+4y^2}\sin(y)$

Explanation

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

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

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

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

Find the differential of the following function:

$h=z\cos(xyz)+y^2\ln(xz^2)$

$(-yz^2\sin(xyz)+\frac{y^2}{x})dx+(-xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z})dz$

$-yz^2\sin(xyz)+\frac{y^2}{x}-xz^2\sin(xyz)+2y\ln(xz^2)+\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z}$

$(yz^2\sin(xyz)+\frac{y^2}{x})dx+(-xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z})dz$

$(-yz^2\sin(xyz)+\frac{y^2}{x})dx+(xz^2\sin(xyz)+2y\ln(xz^2))dy+(\cos(xyz)+xyz\sin(xyz)+\frac{2y^2}{z})dz$

Explanation

The differential of the function is given by

The partial derivatives are

$f_x=-yz^2\sin(xyz)+\frac{y^2}{x}$ , $f_y=-xz^2\sin(xyz)+2y\ln(xz^2)$ , $f_z=\cos(xyz)-xyz\sin(xyz)+\frac{2y^2}{z}$

Differentials

Help Questions

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation

Explanation