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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #931 : Partial Derivatives

Find $f_{yx}$ of the following function:

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

Possible Answers:

$2xyz+x\sec(y)\tan(y)$

$2yz-\sec(y)\tan(y)$

$2yz+\sec(y)\tan(y)$

Correct answer:

$2yz+\sec(y)\tan(y)$

Explanation:

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:

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

The derivative was found using the following rules:

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

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

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

The derivative was found using rules above as well as

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

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Example Question #932 : Partial Derivatives

Find $f_{yx}$ for the following function:

$f(x, y, z)=ze^{xy}$

Possible Answers:

$zx^2e^{xy}$

$zxe^{xy}$

$ze^{xy}+zx^2e^{xy}$

$zxe^{xy}+zx^2e^{xy}$

Correct answer:

$ze^{xy}+zx^2e^{xy}$

Explanation:

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:

$f_y=zxe^{xy}$

The derivative was found using the following rule:

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

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

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

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

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

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Example Question #933 : Partial Derivatives

Find $f_{yz}$ for the function

$f(x,y, z)=xy\cos(z^2)+z^3y$

Possible Answers:

$-x\sin(z^2)+3z^2$

$-2xz\sin(z^2)+3z^2$

$x\cos(z^2)+z^3$

$2xz\sin(z^2)+3z^2$

Correct answer:

$-2xz\sin(z^2)+3z^2$

Explanation:

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:

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

The derivative was found using the following rules:

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

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

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

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

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

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Example Question #934 : Partial Derivatives

Find $f_{y}$ of the function

$f(x, y,z)=z^2y+xe^{y^2}$

Possible Answers:

$2zy+2xye^{y^2}$

$z^2+e^{y^2}+xe^{y^2}$

$z^2+2ye^{y^2}$

$z^2+2xye^{y^2}$

Correct answer:

$z^2+2xye^{y^2}$

Explanation:

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:

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

The derivative was found using the following rules:

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

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Example Question #935 : Partial Derivatives

Find the equation of the plane given the point on the plane (0,2,0) and the normal vector to the plane $n=\left \langle 5,2,1\right \rangle$ .

Possible Answers:

5x+2y+z=4

5x+4y+z=1

2x+5y+z=4

2x+5y=4

Correct answer:

5x+2y+z=4

Explanation:

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

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Example Question #936 : Partial Derivatives

Find $f_{yyz}$ of the following function: y^6z+y^4x^2

Possible Answers:

30y^4

6y^5+4y^3x^2

30y^4z+24yx^2

6y^5z^2+12yx

Correct answer:

30y^4

Explanation:

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

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Example Question #937 : Partial Derivatives

Find f_z of the following function:

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

Possible Answers:

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

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

$-2z\tan(z^2)+xy$

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

Correct answer:

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

Explanation:

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:

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

The derivative was found using the following rules:

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

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Example Question #938 : Partial Derivatives

Find $f_{xxz}$ for the following function:

$f(x, y, z)=x^4yz+\cos(z)\ln(x)$

Possible Answers:

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

$12x^2yz-\frac{\cos(z)}{x^2}$

$12x^2y-\frac{\sin(z)}{x^2}$

$12x^2y+\frac{\sin(z)}{x^2}$

Correct answer:

$12x^2y+\frac{\sin(z)}{x^2}$

Explanation:

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:

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

The following rules were used:

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

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

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

The rules used are stated above.

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

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

The rules we used are stated above, as well as

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

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Example Question #939 : Partial Derivatives

Find f_x of the following function:

$f(x, y, z)=5x\cos(z)+xe^{xy}$

Possible Answers:

$5\sin(z)+e^{xy}+xye^{xy}$

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

$5\cos(z)+xye^{xy}$

$5x\cos(z)+e^{xy}+xye^{xy}$

Correct answer:

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

Explanation:

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

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

and was found using the following rules:

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

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Example Question #940 : Partial Derivatives

Find $f_{xzz}$ of the function

$f(x, y, z)=6xz^2\cos(y)$

Possible Answers:

$6z^2\cos(y)$

$12z\cos(y)$

$12\cos(y)$

Correct answer:

$12\cos(y)$

Explanation:

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:

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

The derivative was found using the following rule:

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

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

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

We used the above rule and the following rule:

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

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

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

The above two rules were used.

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #931 : Partial Derivatives

Example Question #932 : Partial Derivatives

Example Question #933 : Partial Derivatives

Example Question #934 : Partial Derivatives

Example Question #935 : Partial Derivatives

Example Question #936 : Partial Derivatives

Example Question #937 : Partial Derivatives

Example Question #938 : Partial Derivatives

Example Question #939 : Partial Derivatives

Example Question #940 : Partial Derivatives

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