Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #451 : Derivatives

What is the derivative of ?

Possible Answers:

Correct answer:

Explanation:

Recall that when taking the derivative, multiply the exponent by the coefficient in front of the x term and then also subtract one from the exponent:

Recall that the derivative of a constant is just zero.

Example Question #452 : Derivatives

What is the second derivative of ?

Possible Answers:

Correct answer:

Explanation:

Before you can take the second derivative, you must first take the first derivative. Remember to multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent:

Now, take the second derivative:

Example Question #453 : Derivatives

Find the first and second derivative of the function.

Possible Answers:

Correct answer:

Explanation:

To find the first and second derivatives, we use the chain rule.

where .

For the first derivative we have  and  and so

For the second derivative we have  and  and so

Example Question #454 : Derivatives

Find the first and second derivative of the function.

Possible Answers:

Correct answer:

Explanation:

We find the first derivative by deriving term by term. As such,

Taking the second derivative

 

Example Question #455 : Derivatives

What is the first derivative of 

Possible Answers:

Correct answer:

Explanation:

To find the first derivative, look at each term separately. 

For the first term, multiply the exponent by the coefficient () and then subtract one from the exponent to get .

Do the same with the next term:  and then subtract one from the exponent to get .

For the third term, the derivative is just the coefficient (1 in this case). 

The derivative of a constant is 0.

Put those all together to get your answer: .

Example Question #456 : Derivatives

Find the second derivative of .

Possible Answers:

Correct answer:

Explanation:

Before finding the second derivative, you must find the first derivative. Remember to multiply the exponent by the coefficient and then subtract one from the exponent. Evaluate each term separately and then put all together at the end.

The first derivative is: .

Now, take the derivative of the 1st derivative to find the second derivative: 

.

Example Question #457 : Derivatives

Find the first derivative of .

Possible Answers:

Correct answer:

Explanation:

First, chop up the fraction into 2 separate terms and simplify:

Now, take the derivative of that expression. Remember to multiply the exponent by the coefficient and then decrease the exponent by 1:

Example Question #458 : Derivatives

Possible Answers:

Correct answer:

Explanation:

Integrate each term separately. When there is just an x on the denominator, the integral of that is .

Therefore, the first term integrated is:

 

The second term integrated is:

Put those together to get:

Because it's an indefinite integral, remember to add C at the end: 

.

Example Question #5 : Velocity, Speed, Acceleration

Given the velocity function

where  is real number such that , find the acceleration function

.

Possible Answers:

Correct answer:

Explanation:

We can find the acceleration function  from the velocity function by taking the derivative:

We can view the function

as the composition of the following functions

 

so that . This means we use the chain rule

to find the derivative. We have  and , so we have

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