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Calculus 1 : How to find differential functions

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Example Questions

Example Question #411 : Differential Functions

Take the derivative of the function $f(x,y)=x^2ye^{xy^2}$ .

Possible Answers:

$(xye^{xy^2}+x^5e^{xy^2})dx+(y^2e^{xy^2}+2y^5e^{xy^2})dy$

$(y^2e^{xy^2}+2y^5e^{xy^2)}dx+(xye^{xy^2}+x^5e^{xy^2})dy$

$(x^2e^{xy^2}+2x^3y^2e^{xy^2)}dx+(2xye^{xy^2}+x^2y^3e^{xy^2})dy$

$3(xye^{xy^2}+x^5e^{xy^2})dx+2(y^2e^{xy^2}+2y^5e^{xy^2})dy$

$(2xye^{xy^2}+x^2y^3e^{xy^2})dx+(x^2e^{xy^2}+2x^3y^2e^{xy^2)}dy$

Correct answer:

$(2xye^{xy^2}+x^2y^3e^{xy^2})dx+(x^2e^{xy^2}+2x^3y^2e^{xy^2)}dy$

Explanation:

Note that for this problem, we're not told to take the derivative with respect to any particular variable, so it would be prudent to take the derivative with respect to all three. Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: d[e^u]=du(e^u)

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.

For the function $f(x,y)=x^2ye^{xy^2}$

The derivative with respect to x is found to be:

$(2xye^{xy^2}+x^2y^3e^{xy^2})dx$

And the derivative with respect to y is:

$(x^2e^{xy^2}+2x^3y^2e^{xy^2)}dy$

Remember to use the product rule!

The derivative is then the sum of these individual deritavives:

$f'(x,y)=(2xye^{xy^2}+x^2y^3e^{xy^2})dx+(x^2e^{xy^2}+2x^3y^2e^{xy^2)}dy$

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Example Question #411 : Differential Functions

Find the derivative of the following:

$2x^{2}+6+3x^{12}$

Possible Answers:

$4x+6+36x^{11}$

$2x^{2}+3x^{12}$

$4x+36x^{11}$

$x^{2}+x^{12}$

Correct answer:

$4x+36x^{11}$

Explanation:

To find the derivative, simply use the power rule. To use the power rule, multiply the exponent by the coefficient and the result is the new coefficient. Meanwhile, the exponent gets knocked down by one. Do this for each part of the equation, and you will reach the answer:

Apply the power rule:

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

$\frac{d}{dx}3x^{12}=36x^{11}$

Recall that the derivative of a constant equals zero.

$\frac{d}{dx}6=0$

Thus, the answer is:

$4x+36x^{11}$

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Example Question #412 : Differential Functions

Find the dervative of the following:

$5x\cdot7x^{2}$

Possible Answers:

$105x^{3}$

$35x^{2}\cdot70x^{2}$

5+14x

$105x^{2}$

Correct answer:

$105x^{2}$

Explanation:

For this particular problem use the product rule to solve this derivative.

The product rule says that if a=5x and $b=7x^{2}$ , the derivative of ab equals $a\cdot b'+ b\cdot a'$ .

Therefore,

using the power rule which states to multiply the exponent by the coefficient and then decrease the exponent by one, we get the following,

a=5x

$a'=5\cdot 1x^{1-1}=5x^0=5$

and

$b=7x^{2}$

$b'=7\cdot 2x^{2-1}=14x$

Adding these two products together to complete the product rule we get,

$5(7x^{2})+5x(14x)$ .

Now simplify the expression by combining the like terms.

$35x^{2}+70x^{2}$

$105x^{2}$

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Example Question #231 : How To Find Differential Functions

Find the derivative of the following:

$x^{3}+3x+23$

Possible Answers:

$x^{2}+x+24$

$3x^{2}+3$

$x^{2}+x$

Correct answer:

$3x^{2}+3$

Explanation:

Applying the power rule to each term in the function is as follows:

$\frac{d}{dx}x^{3}$ = $3x^{2}$

$\frac{d}{dx}3x$ = 3.

Recall that the derivative of a constant is zero.

Thus, the derivative of 24=0 .

This yields the answer $3x^{2}+3$ .

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Example Question #1451 : Calculus

Find the slope of the tangent line to the following function at point x=2 .

$12x^{3}+3x^{2}+2x+2$

Possible Answers:

200

158

168

170

Correct answer:

158

Explanation:

Applying the power rule to each term in the function is as follows:

$\frac{d}{dx}12x^{3}=36x^{2}$

$\frac{d}{dx}3x^{2}=6x$

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

$\frac{d}{dx}2=0$

The derivative is $36x^{2}+6x+2$ .

Now, substitute "2" for "x".

$36(2^{2})+6(2)+2$

This yields the answer 158.

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Example Question #1452 : Calculus

Find the slope of the tangent line of the following function at x=0 .

$2x^{2}+3x+4$

Possible Answers:

$\frac{2}{3}$

Correct answer:

Explanation:

Applying the power rule to each term in the function is as follows:

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

$\frac{d}{dx}3x=3$

$\frac{d}{dx}4=0$

The derivative of the function is 4x+3 .

Now substitute 0 for x.

4(0)+3

This yields 3 as the slope of the tangent line.

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Example Question #1453 : Calculus

Find the slope of the tangent line of the following function at x=7 .

$x^{3}+3x^{4}+x^{2}-5$

Possible Answers:

4200

4277

5277

2477

Correct answer:

4277

Explanation:

Applying the power rule to each term in the function is as follows:

$\frac{d}{dx}3x^{4}=12x^{3}$

$\frac{d}{dx}x^{3}=3x^{2}$

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

$\frac{d}{dx}(-5)=0$

The derivative is:

$12x^{3}+3x^{2}+2x$ .

Now, substitute 7 for x.

$12(7^{3})+3(7^{2})+2(7)$

This yields 4277 as the answer.

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Example Question #421 : Differential Functions

Find the slope of the tangent line of the following function at x=4 .

$x^{2}+2x+3$

Possible Answers:

Correct answer:

Explanation:

Applying the power rule to each term in the function is as follows:

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

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

$\frac{d}{dx}3=0$

The derivative is: 2x+2

Then, substitute 4 for x.

2(4)+2

This yields the answer 10.

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Example Question #1455 : Calculus

Find the derivative of the following function.

$\frac{8x^{2}}{2}$

Possible Answers:

Correct answer:

Explanation:

First simplify the function to,

$\frac{8x^{2}}{2}=\frac{2\cdot 4x^2}{2}=4x^2$ .

Now to find the derivative, simply use the power rule. To use the power rule, multiply the exponent by the coefficient and the result is the new coefficient. Meanwhile, the exponent gets knocked down by one. Do this for each part of the equation, and you will reach the answer.

Applying the power rule to the function is as follows:

This yields the answer

$4x^2=2\cdot 4x^{2-1}=8x$ .

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Example Question #1456 : Calculus

Find the slope of the tangent line of the following function at x=2.5 .

$6x^{2}+3x+4$

Possible Answers:

Correct answer:

Explanation:

Applying the power rule to each term in the function is as follows:

$\frac{d}{dx}6x^{2}=12x$

$\frac{d}{dx}3x=3$

$\frac{d}{dx}4=0$

The derivative is 12x+3 .

Now, substitute 2.5 for x.

12x+3=12(2.5)+3=33

This yields 33, which is the slope of the tangent line at x=2.5.

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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 #411 : Differential Functions

Example Question #411 : Differential Functions

Example Question #412 : Differential Functions

Example Question #231 : How To Find Differential Functions

Example Question #1451 : Calculus

Example Question #1452 : Calculus

Example Question #1453 : Calculus

Example Question #421 : Differential Functions

Example Question #1455 : Calculus

Example Question #1456 : Calculus

All Calculus 1 Resources

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