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Calculus 1 : Differential Functions

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Example Questions

Example Question #421 : Functions

Find the derivative of the following:

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

Possible Answers:

$x^{2}+x+24$

$x^{2}+x$

$3x^{2}+3$

Correct answer:

$3x^{2}+3$

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.

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 #422 : Functions

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 #421 : Differential Functions

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

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

Possible Answers:

2477

4277

5277

4200

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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Example Question #1453 : Calculus

Find the slope of the tangent line to the following function at x=1 .

$x^{3}+2x^{2}+20$

Possible Answers:

Correct answer:

Explanation:

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

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

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

$\frac{d}{dx}20=0$

The derivative is $3x^{2}+4x$ .

Now, substitute 1 for x.

$3(1)^{2}+4(1)=3+4=7$

This yields 7 as the slope of the tanget line at x=1 .

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Example Question #421 : Functions

Find the derivative of the following function.

$\frac{2x^{2}}{4x}$

Possible Answers:

Does not exist

$-8x^{2}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

To solve this particular problem we will use the quotient rule to find the derivative. The quotient rule states to take the bottom function times the derivative of the top equation, subtract that product from the top function times the derivative of the bottom. Then divide this difference by the bottom function squared.

In mathematical terms,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$

Applying the quotient rule to our function gets,

$\frac{4x(4x)-2x^{2}(4)}{(4x)^{2}}$ .

This simplifies to $\frac{16x^{2}-8x^{2}}{16x^{2}}$ , which further simplifies to $\frac{1}{2}$ .

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Example Question #1455 : Calculus

Find $f_{xyx}$ for the function $f(x,y,z)=xe^{yz}+ysin(xz)$ .

Possible Answers:

-z^2sin(xz)

$ze^{yz}-z^2sin(xz)$

x^2sin(xz)

xy^2sin(xz)

$xze^{yz}$

Correct answer:

-z^2sin(xz)

Explanation:

For this problem, we're asked to take the derivative of a function multiple times, each time with respect to a particular variable.

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

$f(x,y,z)=xe^{yz}+ysin(xz)$

We'll make use of the following derivative properties:

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

Trigonometric derivatives: d[sin(u)]=du(cos(u)); d[cos(u)]=-du(sin(u))

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

Since we're looking for $f_{xyx}$ , begin by taking the derivative with respect to x:

$f_x=e^{yz}+zycos(xz)$

Now take the deritave with respect to y:

$f_{xy}=ze^{yz}+zcos(xz)$

Finally, take the derivative with respect to x once more, and notice how terms without x go to zero:

$f_{xyx}=0-z^2sin(xz)$

$f_{xyx}=-z^2sin(xz)$

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #421 : Functions

Example Question #1451 : Calculus

Example Question #422 : Functions

Example Question #421 : Differential Functions

Example Question #421 : Differential Functions

Example Question #1455 : Calculus

Example Question #1456 : Calculus

Example Question #1453 : Calculus

Example Question #421 : Functions

Example Question #1455 : Calculus

All Calculus 1 Resources

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