Calculus AB : Basic Differentiation Rules

Study concepts, example questions & explanations for Calculus AB

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Example Questions

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Which of the following is the derivative of ?

Possible Answers:

Correct answer:

Explanation:

When finding the derivative of  we use the power rule by multiplying the coefficient of  by the coefficient of .  The form of this looks like .  This is actually a case of using the Chain Rule, but that will be covered in later topics.  So in this case the coefficient of  is  and the coefficient of  is also .  So the .


If we were to find the derivative of , then .

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function 

Possible Answers:

Correct answer:

Explanation:

We use our rule for finding the derivative of .  We know that .  In this case,  and .

 

 

 

Example Question #2 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Which of the following is the correct derivative for ?

 

Possible Answers:

Correct answer:

Explanation:

When we are finding the derivative of  we are finding the rate of change at a certain point/angle of the function .  If we think about finding the derivative using the graph of , we are finding the tangent to  at a certain point/angle.  Thinking about it this way we can see that the derivative of  is given by the  of the given angle.  The general form for this is 

Example Question #3 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function .

Possible Answers:

Correct answer:

Explanation:

We will need to use our rule for finding the derivative of , .

 

 

 

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Which of the following is the correct derivative for ?

Possible Answers:

Correct answer:

Explanation:

To find the derivative of   , we use the rule where  are constants.  We use  because  is the tangent to the graph of  at each given point giving us the rate of change at each given angle of the function .

Example Question #5 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function .

 

Possible Answers:

Correct answer:

Explanation:

We first find the derivative of the quantity of the cosine function.  The derivative of  is .  Now the derivative of  is .  The quantity of the cosine function will stay the same.  So now we use the form   to write our derivative.

 

 

 

Example Question #21 : Basic Differentiation Rules

Which of the following is the correct derivative of ?

Possible Answers:

Correct answer:

Explanation:

The rule for finding the derivative of the natural logarithm is if  then .  So if we wanted to find the derivative of  we would have .

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function .

Possible Answers:

Correct answer:

Explanation:

We know that the derivative of  is  and we also know that when we have to find the derivative of  we keep the exponent the same but multiply the coefficient of  by the coefficient of .

 

 

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Evaluate 

Possible Answers:

Correct answer:

Explanation:

We are able to recognize that this is the definition of derivatives.  Once we identify that this is the definition of derivatives we can see that .  So this question is really just asking us to find .  So our answer here is .

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Which of the following is the derivative of ?

Possible Answers:

Correct answer:

Explanation:

When finding the derivative of  we use the power rule by multiplying the coefficient of  by the coefficient of .  The form of this looks like .  This is actually a case of using the Chain Rule, but that will be covered in later topics.  So in this case the coefficient of  is  and the coefficient of  is also .  So the .


If we were to find the derivative of , then .

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