Other Differential Functions - AP Calculus AB

Card 0 of 12012

Question

Find the derivative.

Answer

Use the power rule to find the derivative.

$\frac{d}{dx}9=0$

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

The derivative is .

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Question

Find the derivative at x=3.

Answer

First, find the derivative using the power rule:

$\frac{d}{dx}6x^3=18x^2$

Now, substitute 3 for x.

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Question

Find the derivative.

$\frac{2}{3x^4}$

Answer

Use the quotient rule to find the derivative.

$\frac{3x^4(0)-2(12x^3)}{(3x^4)^2}=\frac{-24x^3}{9x^8}$

Simplify.

The derivative is $\frac{-8}{3x^5}$ .

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Question

Find the derivative.

$\frac{1}{x^5}$

Answer

Use the quotient rule to find the derivative.

$\frac{x^5(0)-1(5x^4)}{(x^5)^2}=\frac{-5x^4}{x^{10}}=\frac{-5}{x^6}$

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Question

Find the derivative.

$12x^{12}$

Answer

Use the power rule to find the derivative.

$\frac{d}{dx}12x^{12}=144x^{11}$

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Question

Differentiate

Answer

The Chain Rule applies when differentiating compositions of functions. Here, $\small \small f(x)$ equals the composition of two functions, $\small \small x^2$ and $\small \small cos(x)$ . Let $\small \small \small g(x)=x^2$ and $\small \small \small h(x)=cos(x)$ . Then $\small \small f(x)=g(h(x))$ and the Chain Rule tells us to differentiate the outside function $\small \small g(x)$ and multiply the result by the derivative of the inside function $\small \small h(x)$ . In other words, $\small \small f'(x)=g'(h(x))h'(x)$ . Note that the inside function $\small \small h(x)$ is left untouched when the outside function $\small \small g(x)$ is differentiated. Here, $\small \small g'(h(x))=2cos(x)$ and $\small \small h'(x)=-sin(x)$ , so $\small f'(x)=(2cos(x))(-sin(x))$ which simplifies to $\small \small \small f'(x)=-2sin(x)cos(x)$ .

$\small f'(x)=2cos(x)$ is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

$\small f'(x)=2cos(x)-sin(x)$ is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

$\small f'(x)=-2sin(x)$ is a misapplication of the Chain Rule which substitutes the derivative of the inside function for the original inside function rather than multiplying the derivative of the outside function by the derivative of the inside function.

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Question

Differentiate the function

$f(x) = \sin(\cos(2x))$

Answer

To differentiate the function properly, we must use the Chain Rule which is,

$f(x) = h(g(x)) \rightarrow f'(x) = h'(g(x))g'(x)$

Therefore the derivative of the function is,

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

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Question

Differentiate $\small f(x)=sin(6x)$

Answer

The Chain Rule applies when differentiating compositions of functions. Here, $\small \small f(x)$ equals the composition of two functions, $\small sin(x)$ and $\small 6x$ . Let $\small \small \small \small g(x)=sin(x)$ and $\small \small \small \small h(x)=6x$ . Then $\small \small f(x)=g(h(x))$ and the Chain Rule tells us to differentiate the outside function $\small \small g(x)$ and multiply the result by the derivative of the inside function $\small \small h(x)$ . In other words, $\small \small f'(x)=g'(h(x))h'(x)$ . Note that the inside function $\small \small h(x)$ is left untouched when the outside function $\small \small g(x)$ is differentiated. Here, $\small \small \small g'(h(x))=cos(6x)$ and $\small \small \small h'(x)=6$ , so $\small \small f'(x)=(cos(6x))(6)$ which simplifies to $\small \small \small \small f'(x)=6cos(6x)$ .

$\small \small f'(x)=cos(6x)$ is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

$\small \small f'(x)=cos(6x)+6$ is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

$\small \small f'(x)=6cos(x)$ is a misapplication of the Chain Rule which fails to preserve the original inside function when differentiating the outside function.

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Question

Differentiate $\small f(x)=(3x+2)^5$

Answer

The Chain Rule applies when differentiating compositions of functions. Here, $\small \small f(x)$ equals the composition of two functions, $\small x^5$ and $\small 3x+2$ . Let $\small \small \small \small g(x)=x^5$ and $\small \small \small \small h(x)=3x+2$ . Then $\small \small f(x)=g(h(x))$ and the Chain Rule tells us to differentiate the outside function $\small \small g(x)$ and multiply the result by the derivative of the inside function $\small \small h(x)$ . In other words, $\small \small f'(x)=g'(h(x))h'(x)$ . Note that the inside function $\small \small h(x)$ is left untouched when the outside function $\small \small g(x)$ is differentiated. Here, $\small \small \small \small g'(h(x))=5(3x+2)^4$ and $\small \small \small h'(x)=3$ , so $\small \small \small \small f'(x)=(5(3x+2)^4)(3)$ which simplifies to $\small \small \small \small f'(x)=15(3x+2)^4$ .

$\small f'(x)=5(3x+2)^4$ is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

$\small f'(x)=5(3x+2)^4+3$ is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

$\small f'(x)=5(3x+2)^4+3x$ is a misapplication of the Chain Rule which adds the derivative of the outside function to an incorrect derivation of the inside function.

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Question

Find the derivative at .

$2x^{2}-6x$

Answer

Begin by finding the derivative using the power rule.

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

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

The derivative is

Now, substitute for .

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Question

Find the derivative.

$7x^{3}-3$

Answer

Find the derivative using the power rule.

$\frac{d}{dx}7x^3=21x^2$

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

The derivative is $21x^{2}$ .

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Question

Find the derivative at .

$7x^{2}+x-6$

Answer

First, find the derivative using the power rule:

Remember that the power rule is:

$(x^n)'=nx^{n-1}$

Apply this to our problem to get

Now, substitute for .

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Question

Differentiate:

$f(x) = \left(\frac{1}{x}\right)\cos^2(x)$

Answer

To find the derivative of this function we must use the Product Rule and the Chain Rule. First we set

$h(x) = \frac{1}{x}$

and

$g(x) = \cos^2(x)$

Now differentiating both of these functions gives

$h'(x) = \frac{-1}{x^2}$

$g'(x) = 2\cos(x)(-\sin(x)) = -2\cos(x)\sin(x)$

Applying this to the Product Rule gives us,

$f'(x) = \frac{-1}{x^2}\cos^2(x)-\frac{2}{x}\cos(x)\sin(x)$

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Question

Find the derivative.

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

Answer

Use the quotient rule to find this derivative.

Remember that the quotient rule is:

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

Apply this to our problem to get

$\frac{2x(1)-2(x-4)}{4x^2}$

$=\frac{2x-2x-8}{4x^2}$

$=-\frac{8}{4x^2}=-\frac{2}{x^2}$

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Question

Find the derivative at .

Answer

First, use the power rule to find the derivative.

Remember that the power rule is:

$(x^n)'=nx^{n-1}$

Apply this to our problem to get

Now, substitute for .

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Question

Find the derivative.

$x^2\sin(x)$

Answer

Use the product rule to find the derivative.

Remember that the product rule is:

Apply this to our problem to get

$\frac{d}{dx}x^2\sin(x)=x^2\cos(x)+2x\sin (x)$

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Question

Find the derivative.

$\frac{4}{x^3}$

Answer

Use the quotient rule to find this derivative.

Remember that the quotient rule is:

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

Apply this to our problem to get

$\frac{d}{dx}\frac{4}{x^3}=\frac{x^3(0)-2(3x^2)}{x^6}=\frac{-6x^2}{x^6}=\frac{-6}{x^4}$

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Question

Find the derivative.

$-\frac{5}{x^3}$

Answer

Use the quotient rule to find this derivative.

Recall the quotient rule:

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

$h'(x)=\frac{x^3(0)-(-5)3x^2}{x^6}=\frac{15x^2}{x^6}=\frac{15}{x^4}$

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Question

Differentiate

$f(x) = \sin^2(x^2)$

Answer

To differentiate this equation we use the Chain Rule.

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

Using this throughout the equation gives us,

$f'(x) = 2\sin(x^2)\cos(x^2)(2x)$

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Question

Find the derivative of

$f(x) = 4\sin(x+3x^2)$

Answer

To find the derivative of the function we must use the Chain Rule, which is

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

Applying this to the function we get,

$f'(x) = 4\cos(x+3x^2)(1+6x)$

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