Find the derivative using the power rule. 

Remember the power rule:

We can now apply this to our situation.

The derivative is 

First, find the derivative using the power rule. 

Remember the power rule:

We can now apply this to our situation.

The derivative is 

Now, substitute  for .

Use the product rule to find the derivative.

Remember the product rule:

We can now apply this to our situation.

Simplify.

Use the quotient rule to find the derivative. 

Remember the quotient rule:

We can now apply this to our situation.

Find the derivative using the quotient rule. 

Remember the quotient rule:

We can now apply this to our situation.

Find the derivative using the quotient rule. 

Remember the quotient rule:

We can now apply this to our situation.

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.

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.

The solution  validates the mean value theorem by falling within 

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.

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.

Of these two solutions  validates the mean value theorem by valling within 

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.

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.

Both of these solutions validate the mean value theorem by falling within .

Note that for a function that's differentiable on an interval like the one given, there will always be at least one point that satisfies the MVT.

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.

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.

Note that we do not include the start and end points as values that satisfy the mean value theorem. Therefore, there are nine values which fall within the open interval  and satisfy the MVT