Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #251 : Derivatives

Find the derivative of the function

Possible Answers:

None of the other answers

Correct answer:

Explanation:

We could use the chain rule for this function, but we can save some time and work if we recognize that the function can be simplified a bit.

 

Hence

Example Question #252 : Derivatives

Evaluate 

Possible Answers:

None of the other answers

Correct answer:

Explanation:

To evaluate this derivative, we use the Product Rule.

 

. Use the Product Rule. Keep in mind that the derivative of  involves the Chain Rule.

. Factor out an .

Example Question #253 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

and was found using the following rules:

Example Question #254 : Derivatives

What is the second derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

To solce this problem we use the chain rule.

Taking the first derviative we get:

, which simplifies to 

To take the second derivative, we use a combination of the chain and product rules. To use the chain rule on the first term of the equation, we can re-write  as . Taking the second derivative, we get the following:

 

Which simplifies to 

 

 

Which simplifies further to:

Example Question #255 : Derivatives

Find the first 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 #256 : Derivatives

If , find 

Possible Answers:

Correct answer:

Explanation:

By the chain rule:

By the product rule:

Therefore:

Example Question #257 : Derivatives

Find the derivative of .

Possible Answers:

Correct answer:

Explanation:

First, we should simplify the problem by distributing through the parenthesis.

.

Now, since we have a polynomial, we use the power rule to take the derivative.  Multiply the coefficient by the exponent, and reduce the power by 1.

.

Example Question #258 : Derivatives

Find  using implicit differentiation of .

Possible Answers:

Correct answer:

Explanation:

For implicit differentiation, you take a derivative of both the  and  components, and add a  after every  derivative.  Then, using algebra, solve for the  in the equation.

 The derivative is:  , noting that the derivative of a constant equals zero.  Now, we simply rearrange the equation.

Example Question #259 : Derivatives

Find the derivative of .

Possible Answers:

Correct answer:

Explanation:

This derivative has multiple layers of the chain rule.  Whenever woking with a chain rule derivative, always take the derivative of the outside function, leaving the inside function alone.  Then, multiply that by the inside derivative.  Here, our first outside function is .  The derivative of that function is .  Then, we multiply that by the derivatives of the inside (which is another chain rule.  The whole chain looks like this:

.

In the last step, we used the definition   to simplify the answer.

Example Question #260 : Derivatives

Find 

Possible Answers:

Correct answer:

Explanation:

To simplify the problem, it is easiest if we transform the function from in the denominator into the numinator.

.

Now, we just take the derivative using the chain rule:

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