AP Calculus AB : Computation of the Derivative

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #1 : Computation Of The Derivative

The Second Fundamental Theorem of Calculus (FTOC) 

Consider the function equation (1) 

                                          \displaystyle F(x)=\int_a^xf(t)dt                (1)

The Second FTOC states that if: 

  1. \displaystyle f(x) is continuous on an open interval \displaystyle I
  2.  \displaystyle a is in \displaystyle I.
  3. and \displaystyle F(x)  is the anti derivative of \displaystyle f(x) 

 

then we must have,

              \displaystyle F'(x)=f(x)                   (2)

 

Differentiate, 

\displaystyle f(x)=\sin(x^2)+\int_o^{x^2+1}\ln(t+1)dt

 

Possible Answers:

\displaystyle f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]

\displaystyle f'(x)=cos(x^2)-\frac{1}{x^2+2}

\displaystyle f'(x)=2x\cos(x^2)+\ln(x^2+1)

\displaystyle f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]  

\displaystyle f'(x)=2x\left[\cos(x^2)+\frac{1}{x^2+1}\right] 

Correct answer:

\displaystyle f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]

Explanation:

Differentiate: 

\displaystyle f(x)=\sin(x^2)+\int_o^{x^2+1}\ln(t+1)dt


\displaystyle f'(x) = \frac{d}{dx}\sin(x^2)+\frac{d}{dx}\int_o^{x^2+1} \ln(t+1)dt  

 

Both terms must be differentiated using the chain rule. The second term will use a combination of the chain rule and the Second Fundamental Theorem of Calculus. To make the derivative of the second term easier to understand, define a new variable so that the limits of integration will have the form shown in Equation (1) in the pre-question text. 

Let,

\displaystyle u= x^2+1

Therefore, 

\displaystyle \frac{du}{dx}=2x

 

Now we can write the derivative using the chain rule as: 

\displaystyle f'(x) = 2x\cos(x^2)+\frac{du}{dx}\frac{d}{du}\left(\int_o^{u} \ln(t+1)dt\right)

 

 \displaystyle f'(x) = 2x\cos(x^2)+2x\underset{Use_, FTOC}{ \underbrace{\frac{d}{du}\left(\int_o^{u} \ln(t+1)dt\right)}}

 

Let's calculate the derivative with respect to \displaystyle u in the second term using the Second FTOC and then convert back to \displaystyle x

\displaystyle \frac{d}{du}\int_o^{u} \ln(t+1)dt = \ln (u+1) = \ln\left \{ (x^2+1)+1 \right \} = \ln(x^2+2) 

 

Therefore we have, 

\displaystyle f'(x)=2x\cos(x^2)+2x\ln(x^2+2)

 

\displaystyle f'(x)=2x\left[\cos(x^2)+\ln(x^2+2)\right]

Example Question #1 : Derivatives Of Functions

Find the derivative.

\displaystyle 12x^3

Possible Answers:

\displaystyle 0

\displaystyle 36x^2

\displaystyle 24x^2

\displaystyle 36x^3

Correct answer:

\displaystyle 36x^2

Explanation:

Use the power rule to find the derivative.

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

Example Question #2 : Derivatives Of Functions

Find the derivative of the following:

\displaystyle f(x)=3x+2

Possible Answers:

\displaystyle f'(x)=5

\displaystyle f'(x)=3x^2+2x

\displaystyle f'(x)=3

None of the above

Correct answer:

\displaystyle f'(x)=3

Explanation:

To take the derivative you need to bring the power down to the front of the equation, multiplying it by the coefficient and then drop the power.

So:

\displaystyle f(x)=3x+2  becomes \displaystyle f'(x)=3

because the degree of "x" is just one, and once you multiply 3 by 1 you get 3 and drop the power of "x" to 0. The second term is just a constant and the derivative of any term is just 0. 

Example Question #12 : Ap Calculus Ab

Find \displaystyle f'(x):

 

\displaystyle f(x)=\frac{2x}{x}

Possible Answers:

\displaystyle f'(x)=\frac{2x-1}{x}

\displaystyle f'(x)=0

\displaystyle f'(x)=\frac{1}{x}

\displaystyle f'(x)=\frac{2x-1}{x^2}

\displaystyle f'(x)=\frac{1}{x^2}

Correct answer:

\displaystyle f'(x)=0

Explanation:

To do this problem you need to use the quotient rule. So you do

(low)(derivative of the high)-(high)(derivative of the low) all divided by the bottom term squared.

So:

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

Which, when simplified is:

\displaystyle \frac{2x-2x}{x^2}=0

Example Question #1 : Derivatives Of Functions

Find \displaystyle f'(x):

\displaystyle (3x^3)(x)

Possible Answers:

\displaystyle f'(x)=12x^3

\displaystyle f'(x)=9x^3+3x^2

\displaystyle f'(x)=12x^2

\displaystyle f'(x)=9x^2

\displaystyle f'(x)=9x^3

Correct answer:

\displaystyle f'(x)=12x^3

Explanation:

This is the product rule, which is: (derivative of the first)(second)+(derivative of the second)(first)

So:

\displaystyle (9x^2)(x)+(1)(3x^3)=12x^3

Example Question #2 : Derivatives Of Functions

Find the derivative of the following:

\displaystyle f(x)=(x^2)(\frac{1}{x})

Possible Answers:

\displaystyle f'(x)=1

\displaystyle f'(x)=x-\frac{1}{x}

\displaystyle f'(x)=0

\displaystyle f'(x)=x^2-\frac{1}{x^2}

\displaystyle f'(x)=x-1

Correct answer:

\displaystyle f'(x)=1

Explanation:

This is a combination of chain rule and quotient rule.

So:

\displaystyle 2x(\frac{1}{x})+\frac{0-1}{x^2}(x^2)=2-\frac{x^2}{x^2}

Which when simplified you get:

\displaystyle f'(x)=2-1=1

Example Question #3 : Derivatives Of Functions

Find the derivative of the following:

\displaystyle f(x)=sin(x)+tan(x)

Possible Answers:

\displaystyle f'(x)=-cos(x)-sec^2(x)

\displaystyle f'(x)=cos(x)+sec^2(x)

\displaystyle f'(x)=cos(x)-sec^2(x)

\displaystyle f'(x)=-cos(x)+sec^2(x)

Correct answer:

\displaystyle f'(x)=cos(x)+sec^2(x)

Explanation:

This problem is just addition of derivatives using trigonometric functions.

So:

\displaystyle f'(x)=cos(x)+sec^2(x)

 

 

Example Question #1 : Computation Of The Derivative

Find the derivative:

 

\displaystyle \frac{tan(x)}{x^2}

Possible Answers:

\displaystyle f'(x)=\frac{-xsec^2(x)+2tan(x)}{x^3}

\displaystyle f'(x)=\frac{xsec^2(x)-2tan(x)}{x^4}

\displaystyle f'(x)=\frac{-xsec^2(x)-2tan(x)}{x^3}

\displaystyle f'(x)=\frac{xsec^2(x)+2tan(x)}{x^3}

\displaystyle f'(x)=\frac{xsec^2(x)-2tan(x)}{x^3}

Correct answer:

\displaystyle f'(x)=\frac{xsec^2(x)-2tan(x)}{x^3}

Explanation:

The is a quotient rule using a trigonometric function.

So:

\displaystyle f'(x)=\frac{x^2(sec^2(x))-tan(x)(2x)}{x^4}

You can pull out an "x" and cancel it to get:

\displaystyle f'(x)=\frac{xsec^2(x)-2tan(x)}{x^3}

Example Question #5 : Computation Of The Derivative

Find the derivative:

\displaystyle f(x)=x^{-10}

Possible Answers:

\displaystyle f'(x)=\frac{10}{x^{11}}

\displaystyle f'(x)=\frac{-10}{x^9}

\displaystyle f'(x)=\frac{10}{x^9}

\displaystyle f'(x)=\frac{-10}{x^{11}}

Correct answer:

\displaystyle f'(x)=\frac{-10}{x^9}

Explanation:

This is the same concept as a normal derivative just with a negative in the exponent.

\displaystyle f'(x)=-10(x^{-9}) 

which becomes:

\displaystyle f'(x)=\frac{-10}{x^9}

Example Question #6 : Computation Of The Derivative

Calculate \displaystyle f'(x):

\displaystyle \small f'(x)={(x^2)}^2

Possible Answers:

\displaystyle \small f'(x)=2(x^2)(2x)

\displaystyle \small f'(x)=3x^2

\displaystyle \small f'(x)=(x^2)(2x)

\displaystyle \small f'(x)=4x^2

Correct answer:

\displaystyle \small f'(x)=2(x^2)(2x)

Explanation:

This is a power rule that can utilize u-substitution.

So  \displaystyle \small f'(x)=(u)^2*u'

where \displaystyle \small u=x^2

So you get:

\displaystyle \small f'(x)=2u*u'

Plug "u" back in and you get:

\displaystyle \small f'(x)=2(x^2)(2x)

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