Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

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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 #12 : 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 #13 : First And Second Derivatives Of Functions

Find the derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The integral of the function is equal to 

and was found using the following rules:

Note that it is easier to integrate the first term once it is rewritten as .

Example Question #3 : Velocity, Speed, Acceleration

Find the velocity function of the particle if its position is given by the following function:

Possible Answers:

Correct answer:

Explanation:

The velocity function is given by the first derivative of the position function:

and was found using the following rules:

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 #15 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative was found using the following rules:

 

 

Example Question #16 : First And Second Derivatives Of Functions

Find the derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is equal to

 

 

and was found using the following rules:

Example Question #17 : First And Second Derivatives Of Functions

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the function was found using the following rules:

Example Question #18 : First And Second Derivatives Of Functions

Solve for the derivative:  

Possible Answers:

Correct answer:

Explanation:

Write the derivative of cotangent .

The problem requires chain rule since the inner function is .  The chain rule is the derivative of the inner function, and the derivative of  is .

Take the derivative to obtain .  Multiply the value obtained by the use of chain rule.

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