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AP Calculus AB : Computation of the Derivative

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Example Question #37 : Derivatives

Calculate the derivative of the following:

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

Possible Answers:

$\frac{1}{2}x^2$

$\frac{x}{2}$

2x^2

Correct answer:

Explanation:

To find the derivative, use the quotient rule.

The quotient rule requires you to do the following:

$\frac{(Denominator)(Deriv. Numerator)-(Numerator)(Deriv. Denominator)}{(Denominator^2)}$

When you apply it to this problem, you get a final answer of,

$\\\text{Numerator}=x^2 \\\text{Deriv. of Numerator}=2x \\\text{Denominator}=2 \\\text{Deriv. of Denominator}=0$

$\\f'(x)=\frac{2(2x)-x^2(0)}{2^2} \\\\f'(x)=\frac{4x}{4} \\\\f'(x)=x$

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Example Question #38 : Derivatives

Calculate the derivative of the following:

$12x^{5}-4x^4+10x^3$

Possible Answers:

60x^4+16x^3-30x^2

60x^5-16x^4+30x^3

60x^4-16x^3+30x^2

60x^4+16x^3+30x^2

12x^5-4x^3+10x^2

Correct answer:

60x^4-16x^3+30x^2

Explanation:

Use the power rule to multiply the exponent of each term with its coefficient, to get the derivative of each separate term.

Then, decrease the exponent of each term by

Keep all the signs the same, and your final answer will be

$\\f(x)=12x^{5}-4x^4+10x^3 \\f'(x)=5\cdot 12x^{5-1}-4\cdot 4x^{4-1}+3\cdot 10x^{3-1}$

f'(x)=60x^4-16x^3+30x^2

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Example Question #21 : Computation Of The Derivative

Calculate the derivative of the following:

$\sin(x)$

Possible Answers:

$-\sin(x)$

$-\cos(x)$

$-\csc(x)$

$\cos(x)$

$\csc(x)$

Correct answer:

$\cos(x)$

Explanation:

This is a trigonometry identity.

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

The derivative of $\sin(x)$ will always be $\cos(x)$ .

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Example Question #21 : Computation Of The Derivative

Calculate the derivative of the following:

$\cos(x)$

Possible Answers:

$-\sec(x)$

$\sin(x)$

$-\sin(x)$

$\sec(x)$

$-\cos(x)$

Correct answer:

$-\sin(x)$

Explanation:

This is a trigonometry identity.

$\frac{d}{dx}\cos(x)=-\sin(x)$

The derivative of $\cos(x)$ will always be $-\sin(x)$ .

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Example Question #41 : Derivatives

Calculate the derivative of the following:

$x^{\frac{1}{3}}$

Possible Answers:

$-\frac{1}{3\sqrt[3]{x^2}}$

$\frac{1}{3\sqrt[3]{x^2}}$

$\frac{1}{3}x^\frac{-1}{3}$

$\frac{1}{3}x^\frac{1}{3}$

$\frac{1}{3}x$

Correct answer:

$\frac{1}{3\sqrt[3]{x^2}}$

Explanation:

Use the power rule to find the derivative of the function.

In mathematical terms, the power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$

Move the exponent to the front, making it the coefficient.

Next, decrease the exponent by making it $x^{\frac{-2}{3}}$ .

After simplifying, you get

$\frac{1}{3\sqrt[3]{x^2}}$ .

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Example Question #21 : Computation Of The Derivative

Find the derivative of the following equation:

$h(x)=3\sec(x)$

Possible Answers:

$3x\sec(x)\tan(x)$

$3\tan ^{2}(x)$

$3\sec(x)\tan(x)$

$3\sec(3x)\tan(3x)$

None of the other answers.

Correct answer:

$3\sec(x)\tan(x)$

Explanation:

To solve this problem, we need to use the identity that tells us that

$\frac{d}{dx}(\sec (x))=\sec(x)\tan(x)$ .

After using this identity, we tack the 3 back on to get

$h{}'(x)=3\sec(x)\tan(x)$ .

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Example Question #22 : Computation Of The Derivative

x^3 + 3x^2 + 3x -3

Possible Answers:

3x^2-6x+3

3x^2 + 6x +3

3x+3

Correct answer:

3x^2 + 6x +3

Explanation:

To solve this equation we will use power rule.

Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. We will do this for all values in this problem that have an "x" value attached to it.

x^3 = 3x^2 ; 3x^2 = 6x ; 3x=3

The 3 on the end has no "x" so the derivative we will set that equal to just zero. Then combine like terms and express as a single derivative.

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Example Question #44 : Derivatives

Find the derivative.

y = sinx +x^2

Possible Answers:

y'= -sinx - 2x

y' = cosx -2x

y' = -cos x +x^2

y'= sinx + 2x

y' = cosx + 2x

Correct answer:

y' = cosx + 2x

Explanation:

To find the derivative we're going to need to know trig and power rule derivative rules.

Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent. $(nx^{^{n-1}})$

And for trig, the derivative of sinx=cosx .

Therefore using these rules we get: y'= cosx + 2x

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Example Question #45 : Derivatives

Find the derivative of the function.

$y= \frac{x+1}{x-2}$

Possible Answers:

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

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

$\frac{(x-2)-(x+1)}{(x-2)}$

$\frac{(x-2)-(x+1)}{(x+2)}$

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

Correct answer:

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

Explanation:

Quotient Rule states that we take $\frac{f(x)g'(x)-g(x)f'(x)}{(g(x))^2}$

f(x)= x+1 ; f'(x)=1

g(x)=x-2 ; g'(x)=1

Plug this into our formula and we get

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

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Example Question #46 : Derivatives

Find the derivative using chain rule.

y= (x+2)^2

Possible Answers:

2(x+2)^2

2x(x+2)

2(x+2)

Correct answer:

2(x+2)

Explanation:

To find the derivative we need to use chain rule.

Chain Rules states that we work from the outside to the inside. Meaning we will take the derivative of the outside of the equation and multiply it by the derivative of the inside of the equation.

y= (x+2)^2 ;The derivative of the outside will be 2(x+2)^1

And the inside (x+2) derivative will just be 1.

Multiply them together and we are left with

2(x+2)

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AP Calculus AB : Computation of the Derivative

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #37 : Derivatives

Example Question #38 : Derivatives

Example Question #21 : Computation Of The Derivative

Example Question #21 : Computation Of The Derivative

Example Question #41 : Derivatives

Example Question #21 : Computation Of The Derivative

Example Question #22 : Computation Of The Derivative

Example Question #44 : Derivatives

Example Question #45 : Derivatives

Example Question #46 : Derivatives

All AP Calculus AB Resources

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