Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #11 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Example Question #1 : Velocity, Speed, Acceleration

Let 

Find the first and second derivative of the function.

Possible Answers:

Correct answer:

Explanation:

In order to solve for the first and second derivative, we must use the chain rule.

The chain rule states that if

 

and 

then the derivative is

In order to find the first derviative of the function

we set

and

Because the derivative of the exponential function is the exponential function itself, we get

And differentiating  we use the power rule which states

As such

And so

 

To solve for the second derivative we set 

and 

Because the derivative of the exponential function is the exponential function itself, we get

And differentiating  we use the power rule which states

As such

And so the second derivative becomes

 

Example Question #31 : Derivatives

Find the second derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The first derivative of the function is equal to

The second derivative - the derivative of the function above - of the original function is equal to

Both derivatives were found using the following rules:

Example Question #31 : Derivatives

Calculate the derivative of the function given below

Possible Answers:

Correct answer:

Explanation:

We have a function inside of another function, therefore we have a chain-rule.

.

Example Question #32 : Derivatives

In order to find the maximas and minimas of a given function , one must set which quantity equal to zero  and solve for ?

Possible Answers:

Correct answer:

Explanation:

At the location of a maximum or minimum of a function, the tangent line has a slope of zero .  We know the first derivative, , of a function , gives us the value of the tangent line for a given value of .  Therefore, one needs to set the first derivative .

Example Question #33 : Derivatives

Find the slope of the line tangent to , given below, when .

Possible Answers:

Correct answer:

Explanation:

First, we need to find the derivative of , given below as 

.  Plugging a value of  gives us

.

Example Question #34 : Derivatives

Find the derivative of the function , where is any constant.

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is

Notice that this function is constant; it does not contain any variables. ( is stated to be a constant). Hence the derivative is .

Example Question #35 : Derivatives

Find the derivative of the function , where is any constant.

Possible Answers:

None of the other answers

Correct answer:

Explanation:

Looking at the term , since is a constant, the derivative is .

The derivative of uses the Chain Rule. So the derivative is .

Remembering that is just constant, we have the derivative of is , also by the chain rule.

Adding our three results together gives

.

Example Question #36 : Derivatives

Evaluate the derivative of the function , where is any constant.

Possible Answers:

None of the other answers

Correct answer:

Explanation:

It's important to take note that is not a variable, it is a constant. Hence the derivative of is , not . This idea will pop up in many places later on in Calculus 3.

Either the Product Rule, or the Constant Multiple Rule can be used to find the derivative of .

For example, the Product Rule will proceed as follows

.

Example Question #37 : Derivatives

Find  of .

Possible Answers:

Correct answer:

Explanation:

Note that .

For this problem, .  Putting this together with the definition, we arrive at 

.

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