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Calculus 1 : Calculus

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Example Questions

Example Question #602 : Other Differential Functions

The position at a certain point is given by:

$f(x)=\frac{1}{2}x^{2} - 4x + 5$

What is the velocity at x=5 ?

Possible Answers:

Correct answer:

Explanation:

In order to find the velocity of a given point, you must first find the derivative of the position function.

In this case, the derivative of the position function is:

f'(x)=x-4

Then, find f'(x) when x=5 .

The answer is:

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Example Question #603 : Other Differential Functions

y=(13x^3+x)^4

Find .

Possible Answers:

y'=4(39x^2+1)^3

y'=4(39x^2+1)

y'=(13x^3+x)^3(39x^2+1)^4

y'=4(13x^3+x)^3(39x^2+1)

Correct answer:

y'=4(13x^3+x)^3(39x^2+1)

Explanation:

We can solve this problem using the chain rule:

Let f(x)=z^4 and g(x)=13x^3+x

We want to find $f'(g(x))\cdot g'(x)$

g'(x)=39x^2+1

and

f'(x)=4x^3

So,

y'=4(13x^3+x)^3(39x^2+1) .

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Example Question #604 : Other Differential Functions

$y=(3x^2-5x)\cdot(4x^4-9)$

Find .

Possible Answers:

y'=96x^4-80x^3

y'=24x^5-20x^4-54x+45

y'=72x^5-100x^4-54x+45

y'=48x^5-80x^4-54x+45

Correct answer:

y'=72x^5-100x^4-54x+45

Explanation:

Multiplying the polynomials together and taking the derivative of each term will yield the correct function for y'.

$y=(3x^2-5x)\cdot(4x^4-9)$

=12x^6-20x^5-27x^2+45x

Using the power rule for each term:

The power rule states,

$f(x)=x^n\rightarrow f'(x)=nx^{n-1}$ .

Therefore the derivative becomes,

y'=72x^5-100x^4-54x+45 .

We could also find the answer through the product rule:

$f(x)=g(x)\cdot h(x)$

f'(x)=g(x)h'(x) + h(x)g'(x)

g(x)=3x^2-5x

g'(x)=6x-5

h(x)=4x^4-9

h'(x)=16x^3

f'(x)=(3x^2-5x)(16x^3)+(4x^4-9)(6x-5)

=48x^5+24x^5-80x^4-20x^4-54x +45

=72x^5-100x^4-54x+45

Notice that the product rule in this case is not more efficient, since you still have to FOIL out h(x)g'(x), and then group like terms together. It is important to recognize when using the product rule is an asset and when it is more complicated than necessary.

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Example Question #605 : Other Differential Functions

f(x)=6x^6+30x+1009

Find f''(2) .

Possible Answers:

f''(x)=1152

f''(2)=2880

f''(x)=0

f''(x)=19080

Correct answer:

f''(2)=2880

Explanation:

To find the answer, first find the first derivative, f''(x).

To find the derivative use the power rule which states,

$f(x)=x^n\rightarrow f'(x)=nx^{n-1}$ .

Thus we get the following.

f(x)=6x^6+30x+1009

f'(x)=36x^5+30

f''(x)=180x^4

Plugging in 2 for x, we find:

f''(2)=180(2)^4

=180 (16)

=2880

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Example Question #606 : Other Differential Functions

Given,

p=5x^3+2x

q=13x^4+4x^2-1

What is the differential function of y=p-q ?

Possible Answers:

y'=52x^3-15x^2+8x+2

y'=-13x^4+11x^2+3

y'=-52x^3+15x^2-8x+2

y'=47x^3+6x

Correct answer:

y'=-52x^3+15x^2-8x+2

Explanation:

When finding the derivative of functions combined with addition or subtraction, we can simply combine the functions with the appropriate operation (subtraction in this case) and take the derivative of each term in the resulting function.

p-q=(5x^3+2x)-(13x^4+4x^2-1)

=-13x^4+5x^3-4x^2+2x-1

When we take the derivative of the difference we get:

y'=-52x^3+15x^2-8x+2

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Example Question #607 : Other Differential Functions

$f(x) = x^{2} + 3x - 4sinx$

Find the derivative.

Possible Answers:

f'(x) = x + 3 - cosx

f'(x) = 2x + 3 - 4cosx

$f'(x) = 2x^{2} + 3x - 4cosx$

f'(x) = 2x + 3x - 4cosx

$f'(x) = 2x^{2} + 3x^{2} - 4(\cos x)^{2}$

Correct answer:

f'(x) = 2x + 3 - 4cosx

Explanation:

When differentiating a function, you must understand how to find the derivative of different types of functions:

If f(x) = c (constant), then f'(x) = 0 .

If $f(x) = x^{n}$ , then $f'(x) = (n)x^{n-1}$ .

If $f(x) = \sin x$ , then $f'(x) = \cos x$ .

There are multiple different rules that apply when differentiating different kinds of functions.

In this case, you find the derivative of each portion of the function:

$f(x) = x^{2} + 3x - 4sinx$ $\large \huge \rightarrow$ $\huge f'(x) = (2)x^{(2-1)} + (1)3x^{(1-1)} - 4cosx$

And so the simplified answer is:

f'(x) = 2x + 3 - 4cosx

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Example Question #791 : Differential Functions

$f(x) = 2x^{5} + 3e^{x} - 45$

Differentiate the function.

Possible Answers:

$f'(x) = 10x^{4} + 3e^{x}$

$f'(x) = 5x^{4 }+ 3e^{x }- 45$

$f'(x) = 5x^{4} + 3e^{x}$

$f'(x) = 10x^{4} + 3e^{x} - 45$

$f'(x) = 10x^{5}+ 3e^{x}$

Correct answer:

$f'(x) = 10x^{4} + 3e^{x}$

Explanation:

When differentiating a function, you must understand how to find the derivative of different types of functions:

If f(x) = c (constant), then f'(x) = 0 .

If $\large f(x) = e^{}x$ , then $\large f'(x) = e^{x}$ ^.

If $f(x) = x^{n}$ , then $f'(x) = (n)x^{n-1}$ .

If $f(x) = \sin x$ , then $f'(x) = \cos x$ .

There are multiple different rules that apply when differentiating different kinds of functions.

In this case, you find the derivative of each portion of the function:

$f(x) = 2x^{5} + 3e^{x} - 45$ $\large \rightarrow$ $f'(x) = (5)2x^{(5-1)} + 3e^{x} - 0$

And so the simplified answer is:

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Example Question #792 : Differential Functions

$f(x) = 12\cos x - 3\sin x + 2x$

Differentiate the function.

Possible Answers:

$f'(x) = -12\sin x - 3\cos x$

$f'(x) = -12\cos x - 3 \cos x + x$

$f'(x) = -12\cos x - 3\sin x + 2$

$f'(x) = -12\sin x - 3\cos x + 2$

$f'(x) = 12\sin x - 3\cos x + 2$

Correct answer:

$f'(x) = -12\sin x - 3\cos x + 2$

Explanation:

When differentiating a function, you must understand how to find the derivative of different types of functions:

If f(x) = c (constant), then f'(x) = 0 .

If $\large f(x) = e^{}x$ , then $\large f'(x) = e^{x}$ ^.

If $f(x) = x^{n}$ , then $f'(x) = (n)x^{n-1}$ .

If $f(x) = \sin x$ , then $f'(x) = \cos x$ .

There are multiple different rules that apply when differentiating different kinds of functions.

In this case, you find the derivative of each portion of the function:

$f(x) = 12\cos x - 3\sin x + 2x$ $\LARGE \rightarrow$ $f'(x) = 12(-\sin x) - 3(\cos x) + (1)2x^{(1-1)}$

And so the simplified answer is: $f'(x) = -12\sin x - 3\cos x + 2$

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Example Question #1821 : Calculus

Find the second derivative of f(x)=-2x^4+6x^3-x^2+5x-1 .

Possible Answers:

$f{}''(x)=-4x^2+36x-2$

$f{}''(x)=-24x^2+36x$

$f{}''(x)=-24x^2-36x+2$

$f{}''(x)=-24x^2+36x-2$

Correct answer:

$f{}''(x)=-24x^2+36x-2$

Explanation:

In order to find the second derivative, you must first find the first derivative. To find the derivative, multiply the exponent by the leading coefficient and then subtract 1 from the exponent. Therefore, the first derivative is: $f{}'(x)=-8x^3+18x^2-2x+5$ . Then, take the derivative of the first derivative to find your second derivative, which is: $f{}''(x)=-24x^2+36x-2$ .

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Example Question #612 : How To Find Differential Functions

Find $\frac{dy}{dx}$ for the function below:

y^2 + y +x =0

Hint: Use implicit differentiation.

Possible Answers:

$\frac{dy}{dx}={2y+x+1}$

$\frac{dy}{dx}= x +2$

$\frac{dy}{dx}=\frac{-1}{2x+1}$

$\frac{dy}{dx}=\frac{-1}{2y+1}$

$\frac{dy}{dx}= y^2+1$

Correct answer:

$\frac{dy}{dx}=\frac{-1}{2y+1}$

Explanation:

Take the derivative of each term. Remember that the derivative of y should be dy/dx.

$\frac{d(y^2 + y +x =0)}{dx} = 2y\frac{dy}{dx}+\frac{dy}{dx}+1=0$

Remember to follow through with the chain rule for y^2 . Have to multiply by the derivative of the inside of the squared, which is just $\frac{dy}{dx}$ .

Now isolate $\frac{dy}{dx}$ :

$\frac{dy}{dx}(2y+1)=-1$

$\frac{dy}{dx}=\frac{-1}{2y+1}$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #602 : Other Differential Functions

Example Question #603 : Other Differential Functions

Example Question #604 : Other Differential Functions

Example Question #605 : Other Differential Functions

Example Question #606 : Other Differential Functions

Example Question #607 : Other Differential Functions

Example Question #791 : Differential Functions

Example Question #792 : Differential Functions

Example Question #1821 : Calculus

Example Question #612 : How To Find Differential Functions

All Calculus 1 Resources

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