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Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #321 : Other Differential Functions

Find the derivative at x=6 .

$2x^{2}-6x$

Possible Answers:

Correct answer:

Explanation:

Begin by finding the derivative using the power rule.

$\frac{d}{dx}2x^{2}=4x$

$\frac{d}{dx}-6x=-6$

The derivative is 4x-6.

Now, substitute for .

4(6)-6=24-6=18

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Example Question #321 : Other Differential Functions

Find the slope of the function f(x,y)=(x^2+y)^3 at the point (2,5) .

Possible Answers:

(3,1)

(243,81)

(972,243)

(9,3)

(81,27)

Correct answer:

(972,243)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y)=(x^2+y)^3 at the point (2,5)

$\frac{\delta f}{\delta x}=6x(x^2+y)^2;6(2)(2^2+5)^2=972$

$\frac{\delta f}{\delta y}=3(x^2+y)^2;3(2^2+5)^2=243$

The slope is (972,243)

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Example Question #323 : Other Differential Functions

Find the slope of the function f(x,y,z)=1^x+2^y+3^z at the point (3,2,1) .

Possible Answers:

(1,4,3)

(0,4.4,6.6)

(0,2.8,3.3)

(1.1,2.2,3.3)

(8.1,0,9.7)

Correct answer:

(0,2.8,3.3)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y,z)=1^x+2^y+3^z at the point

$\frac{\delta f}{\delta x}=1^xln(1);1^3ln(1)=0$

$\frac{\delta f}{\delta y}=2^yln(2);2^2ln(2)=2.8$

$\frac{\delta f}{\delta z}=3^zln(3);3^1ln(3)=3.3$

The slope is (0,2.8,3.3)

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Example Question #1533 : Calculus

Find the slope of the function $f(x,y,z)=2^{(x+y+z)^2}$ at the point (0,1,2) .

Possible Answers:

(603.3,603.3,603.3)

(903.3,602.2,301.1)

(2129.3,2129.3,2129.3)

(212.3,424.6,636.9)

(709.8,1419.5,2129.3)

Correct answer:

(2129.3,2129.3,2129.3)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

d[a^u]=a^uduln(a)

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y,z)=2^{(x+y+z)^2}$ at the point (0,1,2)

$\frac{\delta f}{\delta x}=2(x+y+z)2^{(x+y+z)^2}ln(2);2(0+1+2)2^{(0+1+2)^2}ln(2)=2129.3$

$\frac{\delta f}{\delta y}=2(x+y+z)2^{(x+y+z)^2}ln(2);2(0+1+2)2^{(0+1+2)^2}ln(2)=2129.3$

$\frac{\delta f}{\delta z}=2(x+y+z)2^{(x+y+z)^2}ln(2);2(0+1+2)2^{(0+1+2)^2}ln(2)=2129.3$

The slope is (2129.3,2129.3,2129.3)

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Example Question #322 : Other Differential Functions

Find the slope of the function f(x,y,z)=z(x^2+y)^2 at point (1,2,3) .

Possible Answers:

(4,12,36)

(9,9,9)

(27,9,3)

(36,18,9)

(12,36,108)

Correct answer:

(36,18,9)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y,z)=z(x^2+y)^2 at point (1,2,3)

$\frac{\delta f}{\delta x}=4xz(x^2+y);4(1)(3)(1^2+2)=36$

$\frac{\delta f}{\delta y}=2z(x^2+y);2(3)(1^2+2)=18$

$\frac{\delta f}{\delta z}=(x^2+y)^2;(1^2+2)^2=9$

The slope is (36,18,9)

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Example Question #513 : Functions

Find the slope of the function f(x,y)=sin(2^yx) at the point (2,3) .

Possible Answers:

(-2.33,4.56)

(3.92,5.15)

(-8.88,-17.17)

(-7.66,-10.62)

(-1.78,-9.02)

Correct answer:

(-7.66,-10.62)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Trigonometric derivative:

d[sin(u)]=cos(u)du

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y)=sin(2^yx) at the point (2,3)

$\frac{\delta f}{\delta x}=2^{y}cos(2^{y}x);2^{3}cos(2^{3}(2))=-7.66$

$\frac{\delta f}{\delta y}=2^{y}xln(2)cos(2^{y}x);2^{3}(2)ln(2)cos(2^{3}(2))=-10.62$

The slope is (-7.66,-10.62)

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Example Question #514 : Functions

Find the slope of the function $f(x,y)=3^{xln(y^2)}$ at the point (1,3) .

Possible Answers:

(26.98,8.19)

(7.28,32.11)

(32.11,7.28)

(8.19,26.98)

(1.25,8.67)

Correct answer:

(26.98,8.19)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y)=3^{xln(y^2)}$ at the point (1,3)

$\frac{\delta f}{\delta x}=ln(y^2)3^{xln(y^2)}ln(3);ln(3^2)3^{ln(3^2)}ln(3)=26.98$

$\frac{\delta f}{\delta y}=\frac{2x(3^{xln(y^2)})ln(3)}{y};\frac{2(3^{ln(3^2)})ln(3)}{3}=8.19$

The slope is (26.98,8.19)

Report an Error

Example Question #323 : Other Differential Functions

Find the slope of the function $f(x,y)=(x+\frac{1}{y})^2$ at the point (2,4) .

Possible Answers:

(7.75,-0.32)

(9,-32)

(4.5,-0.28)

(3,-6)

(1,-12.5)

Correct answer:

(4.5,-0.28)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y)=(x+\frac{1}{y})^2$ at the point (2,4)

$\frac{\delta f}{\delta x}=2(x+\frac{1}{y});2(2+\frac{1}{4})=4.5$

$\frac{\delta f}{\delta y}=-\frac{2}{y^2}(x+\frac{1}{y});-\frac{2}{4^2}(2+\frac{1}{4})=-0.28$

The slope is (4.5,-0.28)

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Example Question #516 : Functions

Find the slope of the function $f(x,y)=x^2(3^{xy^2})$ at the point (2,1) .

Possible Answers:

(11.8,79.3)

(75.6,158.2)

(14.9,32.3)

(22.7,98.3)

(152.2,7.8)

Correct answer:

(75.6,158.2)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential:

d[a^u]=a^uduln(a)

Product rule: d[uv]=udv+vdu

Note that u and v may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y)=x^2(3^{xy^2})$ at the point (2,1)

${\frac{\delta f}{\delta x}=2x(3^{xy^2})+x^2y^2(3^{xy^2})ln(3);2(2)(3^{(2)(1)^2})+(2)^2(1)^2(3^{(2)(1)^2})ln(3)=75.6$

$\frac{\delta f}{\delta y}=2x^3y(3^{xy^2})ln(3);2(2)^3(1)(3^{(2)(1)^2})ln(3)=158.2$

The slope is (75.6,158.2)

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Example Question #517 : Functions

Find the slope of the function f(x,y,z)=(x+y)^2(x+z^2) at the point (1,2,3) .

Possible Answers:

(42,24,9)

(14,8,3)

(69,60,54)

(9,27,54)

(23,20,18)

Correct answer:

(69,60,54)

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Product rule: d[uv]=udv+vdu

Note that u and v may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of f(x,y,z)=(x+y)^2(x+z^2) at the point (1,2,3)

$\frac{\delta f}{\delta x}=2(x+y)(x+z^2)+(x+y)^2;2(1+2)(1+3^2)+(1+2)^2=69$

$\frac{\delta f}{\delta y}=2(x+y)(x+z^2);2(1+2)(1+3^2)=60$

$\frac{\delta f}{\delta z}=2z(x+y)^2;2(3)(1+2)^2=54$

The slope is (69,60,54)

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Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #321 : Other Differential Functions

Example Question #321 : Other Differential Functions

Example Question #323 : Other Differential Functions

Example Question #1533 : Calculus

Example Question #322 : Other Differential Functions

Example Question #513 : Functions

Example Question #514 : Functions

Example Question #323 : Other Differential Functions

Example Question #516 : Functions

Example Question #517 : Functions

All Calculus 1 Resources

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