Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

varsity tutors app store varsity tutors android store

Example Questions

Example Question #901 : Differential Functions

Find the first derivative of 

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We need to differentiate term by term, applying the power rule,

This gives us

 

Example Question #902 : Differential Functions

Find the derivative. 

Possible Answers:

Correct answer:

Explanation:

Use the power rule to find this solution.

Therefore, the derivative is 

Example Question #903 : Differential Functions

Find the derivative.

Possible Answers:

Correct answer:

Explanation:

The derivative of a constant is always 0.

Example Question #904 : Differential Functions

Find the derivative.

 

Possible Answers:

Correct answer:

Explanation:

Use the power rule to find the derivative.

Example Question #905 : Differential Functions

Find the derivative. 

Possible Answers:

Correct answer:

Explanation:

Use the power rule to find the derivative.

Thus, the derivative is 

Example Question #906 : Differential Functions

Find the derivative.

Possible Answers:

Correct answer:

Explanation:

Use the power rule to find this derivative.

Thus, the derivative is 

Example Question #907 : Differential Functions

Find the derivative. 

Possible Answers:

Correct answer:

Explanation:

Recall that the derivative of a constant is zero. 

Example Question #908 : Differential Functions

What is the derivative of ?

Possible Answers:

Correct answer:

Explanation:

When taking the derivative, remember to multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent. Therefore, after differentiating each term, you get: .

Example Question #2 : The Mean Value Theorem

Let  on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.

Possible Answers:

Correct answer:

Explanation:

The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval  for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

Meanvaluetheorem

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of  on the interval 

Then take the difference of the two and divide by the interval.

Now find the derivative of the function; this will be solved for the value(s) found above.

 

Derivative of an exponential: 

Derivative of a natural log: 

Product rule: 

Using a calculator, we find the solution , which fits within the interval , satisfying the mean value theorem.

Example Question #909 : Differential Functions

Find the derivative. 

Possible Answers:

Correct answer:

Explanation:

Use the power rule to find the derivative. 

Thus, the derivative is 

Learning Tools by Varsity Tutors