Calculus 2 : Derivative Review

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #161 : First And Second Derivatives Of Functions

Find the first derivative of 

Possible Answers:

Correct answer:

Explanation:

Step 1: Take derivative of the first term...

 becomes 

Step 2: Take derivative of the second term....
 becomes 
Step 3: Take the derivative of the third term....

 becomes 
Step 4: Take the derivative of the fourth term...

 becomes 
Step 5: Rearrange the resulting terms from highest degree to lowest degree. 
The degree of a term is the exponent.
So, from highest to lowest, we have 6,3,2.

The answer is: 

Example Question #361 : Derivative Review

Compute the first derivative of the function:

Possible Answers:

Correct answer:

Explanation:

This function is a polynomial made up of powers of x; the power rule for derivatives is:

For our equation:

Example Question #162 : First And Second Derivatives Of Functions

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The rule for derivatives of exponential is:

Keeping in mind the chain rule, our derivative is:

Note that because e5 is a constant, its derivative is zero.

Example Question #163 : First And Second Derivatives Of Functions

Compute the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative rule for natural log functions is:

Keeping in mind the chain rule, the derivative is then:

Example Question #164 : First And Second Derivatives Of Functions

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

This derivative requires use of the chain rule:

First, taking the derivative of the tangent:

Next, the derivative of the power of e on the inside of the parenthesis:

Then, the derivative of the square root of x in the power:

Combining the terms into one fraction, this simplifies to:

Example Question #362 : Derivative Review

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

This derivative is calculated with use if the chain rule:

First, we take the derivative of natural log:

Then, we multiply by the derivative of the term inside the natural log:

Finally, this is multiplied by the derivative of the term inside the cosine:

Then, we simplify and make use of the definition of cotangent;

Example Question #361 : Derivative Review

Compute the second derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

To find the second derivative of the function

First, we have to take the second derivative:

Then, we take the derivative of f'(x) to get the second derivative of f(x):

 

The only derivative rule needed here is the power rule:

Example Question #1493 : Calculus Ii

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The quotient rule for derivatives is:

Where in our case:

and the derivatives of these functions are:

Applying the quotient rule to these functions, we get:

Factoring out a four from both the numerator and denominator we can simplify this expression to get our final answer:

Example Question #361 : Derivative Review

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

he quotient rule for derivatives is:

Where in our case:

and the derivatives of these functions are:

Applying the quotient rule to our function, we get:

Simplified, we get:

 

Example Question #1495 : Calculus Ii

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

This problem requires a combination of the quotient and chain rules:

The quotient rule for derivatives is:

Where in our case:

and the derivatives of these functions are:

Applying the quotient rule to these functions, we get:

Simplifying this expression, we get:

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