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Calculus 1 : Differential Functions

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Example Questions

Example Question #1664 : Calculus

Find the slope of the function f(x,y)=cos(x)tan(y) at $\left(\frac{\pi}{4},\frac{\pi}{4}\right)$ .

Possible Answers:

$(\sqrt2,-\sqrt{2})$

$\left(-\frac{\sqrt2}{2},\sqrt{2}\right)$

$\left(-\sqrt2,\frac{\sqrt{2}}{2}\right)$

$\left(\frac{\sqrt2}{2},-\sqrt{2}\right)$

$(-\sqrt2,\sqrt{2})$

Correct answer:

$\left(-\frac{\sqrt2}{2},\sqrt{2}\right)$

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:

Trigonometric derivative:

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

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

f(x,y)=cos(x)tan(y) at $(\frac{\pi}{4},\frac{\pi}{4})$

$\frac{\delta f}{\delta x}=-sin(x)tan(y);-sin(\frac{\pi}{4})tan(\frac{\pi}{4})=-\frac{\sqrt2}{2}$

$\frac{\delta f}{\delta y}=\frac{cos(x)}{cos^2(y)};\frac{cos(\frac{\pi}{4})}{cos^2(\frac{\pi}{4})}=\sqrt{2}$

The slope is $(-\frac{\sqrt2}{2},\sqrt{2})$

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Example Question #451 : Other Differential Functions

Find the slope of the function f(x,y)=ytan(x) at (2,2) .

Possible Answers:

(4.8,0.9)

(11.5,-2.2)

(-2.2,11.5)

(0.9,4.8)

(13.1,-2.2)

Correct answer:

(11.5,-2.2)

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:

Trigonometric derivative:

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

f(x,y)=ytan(x) at (2,2)

$\frac{\delta f}{\delta x}=\frac{y}{cos^2(x)};\frac{2}{cos^2(2)}=11.5$

$\frac{\delta f}{\delta y}=tan(x);tan(2)=-2.2$

The slope is (11.5,-2.2)

Report an Error

Example Question #1672 : Calculus

Find the slope of the function f(x,y)=tan(x^y) at (3,3) .

Possible Answers:

(111.7,123.3)

(419.2,397.3)

(211.8,244.9)

(352.7,128.0)

(316.4,347.6)

Correct answer:

(316.4,347.6)

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:

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

f(x,y)=tan(x^y) at (3,3)

$\frac{\delta f}{\delta x}=\frac{yx^{y-1}}{cos^2(x^y)};\frac{(3)(3^{3-1})}{cos^2(3^3)}=316.4$

$\frac{\delta f}{\delta y}=\frac{x^{y}ln(x)}{cos^2(x^y)};\frac{3^{3}ln(3)}{cos^2(3^3)}=347.6$

The slope is

(316.4,347.6)

Report an Error

Example Question #1673 : Calculus

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

Possible Answers:

(-0.065,-0.641)

(0.641,0.065)

(0.641,-0.065)

(0.065,0.641)

(-0.065,0.641)

Correct answer:

(-0.065,0.641)

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:

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

$f(x,y)=x^{tan(y)}$ at (2,3)

$\frac{\delta f}{\delta x}=tan(y)(x^{tan(y)-1});tan(3)(2^{tan(3)-1})=-0.065$

$\frac{\delta f}{\delta y}=\frac{x^{tan(y)}ln(x)}{cos^2(y)};\frac{2^{tan(3)}ln(2)}{cos^2(3)}=0.641$

The slope is (-0.065,0.641)

Report an Error

Example Question #1674 : Calculus

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

Possible Answers:

(0.46,4.84)

(-0.46,4.84)

(4.84,0.46)

(-0.46,-4.84)

(-4.84,-0.46)

Correct answer:

(-4.84,-0.46)

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:

Trigonometric derivative:

$d[tan(u)]=\frac{du}{cos^2(u)}$

Quotient rule: $d[\frac{u}{v}]=\frac{vdu-udv}{v^2}$

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.

$f(x,y)=\frac{y}{tan(x)}$ at (2,4)

$\frac{\delta f}{\delta x}=\frac{-\frac{y}{cos^2(x)}}{tan^2(x)}=\frac{-y}{sin^2(x)};\frac{-4}{sin^2(2)}=-4.84$

$\frac{\delta f}{\delta y}=\frac{1}{tan(x)};\frac{1}{tan(2)}=-0.46$

The slope is (-4.84,-0.46)

Report an Error

Example Question #451 : How To Find Differential Functions

Determine the slope of the function $f(x,y)=3^{xtan(y)}$ at (1,4) .

Possible Answers:

(2.41,5.68)

(2.22,10.31)

(1.45,2.45)

(4.54,9.17)

(1.90,6.92)

Correct answer:

(4.54,9.17)

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:

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

$f(x,y)=3^{xtan(y)}$ at (1,4)

$\frac{\delta f}{\delta x}=tan(y)(3^{xtan(y)})ln(3);tan(4)(3^{1tan(4)})ln(3)=4.54$

$\frac{\delta f}{\delta y}=\frac{x(3^{xtan(y)})ln(3)}{cos^2(y)};\frac{1(3^{1tan(4)})ln(3)}{cos^2(4)}=9.17$

The slope is (4.54,9.17)

Report an Error

Example Question #1676 : Calculus

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

Possible Answers:

(-0.133,25.671)

(0.808,132.89)

(0.069,-54.582)

(0.011,96.224)

(-0.019,-84.025)

Correct answer:

(-0.019,-84.025)

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:

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

f(x,y)=tan^x(y) at (2,5)

$\frac{\delta f}{\delta x}=tan^x(y)ln(tan(y));tan^2(5)ln(tan(5))=-0.019$

$\frac{\delta f}{\delta y}=\frac{xtan^{x-1}(y)}{cos^2(y)};\frac{2tan^{2-1}(5)}{cos^2(5)}=-84.025$

The slope is (-0.019,-84.025)

Report an Error

Example Question #461 : Other Differential Functions

Find the slope of the function f(x,y)=3x^2y+2xy+4xy^2 at (1,1) .

Possible Answers:

(13,12)

(13,13)

(9,9)

(12,13)

(12,12)

Correct answer:

(12,13)

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.

f(x,y)=3x^2y+2xy+4xy^2 at (1,1)

$\frac{\delta f}{\delta x}=6xy+2y+4y^2;6(1)(1)+2(1)+4(1)^2=12$

$\frac{\delta f}{\delta y}=3x^2+2x+8xy;3(1)^2+2(1)+8(1)(1)=13$

The slope is (12,13)

Report an Error

Example Question #462 : Other Differential Functions

Find the slope of the function f(x,y)=4x^3+3x^2y+2xy^2+y^3 at (1,1) .

Possible Answers:

(20,20)

(20,10)

(15,15)

(10,20)

(10,10)

Correct answer:

(20,10)

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:

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.

f(x,y)=4x^3+3x^2y+2xy^2+y^3 at (1,1)

$\frac{\delta f}{\delta x}=12x^2+6xy+2y^2;12(1)^2+6(1)(1)+2(1)^2=20$

$\frac{\delta f}{\delta y}=3x^2+4xy+3y^3;3(1)^2+4(1)(1)+3(1)^3=10$

The slope is (20,10)

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Example Question #1679 : Calculus

Find the slope of the function f(x,y)=xtan(y)+ycos(x) at $(\pi,\pi)$ .

Possible Answers:

$(\pi+1,0)$

$(1-\pi,0)$

$(0,\pi)$

$(0,\pi-1)$

$(\pi+1,\pi)$

Correct answer:

$(0,\pi-1)$

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:

Trigonometric derivative:

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

$d[tan(u)]=\frac{du}{cos^2(u)}$

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.

f(x,y)=xtan(y)+ycos(x) at $(\pi,\pi)$

$\frac{\delta f}{\delta x}=tan(y)-ysin(x);tan(\pi)-\pi sin(\pi)=0$

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

The slope is $(0,\pi-1)$

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1664 : Calculus

Example Question #451 : Other Differential Functions

Example Question #1672 : Calculus

Example Question #1673 : Calculus

Example Question #1674 : Calculus

Example Question #451 : How To Find Differential Functions

Example Question #1676 : Calculus

Example Question #461 : Other Differential Functions

Example Question #462 : Other Differential Functions

Example Question #1679 : Calculus

All Calculus 1 Resources

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