This function is just a function inside of a function. This means we have to use the chain rule. The chain rule states that the derivative of .

The derivative of is and the derivative of is . This makes the derivative

This makes sense because .

The points where the derivative of the function is equal to 0 are called critical points. They are either local maximums, local minimums, or do not exist. The derivative of is . The derivative of the function is

.

We must now set it equal to zero and factor to solve.

Now we must plug in points to the left and right of the critical points into the derivative function to find the local min.

This means the function is increasing until it hits x=2, then it decreases until it hits x=4 and begins increasing again. This makes x=4 the local minumum

To find the minimum of a function, start by finding the critical points of that function, or points where the derivative is equal to zero. Use the power rule to find the derivative:

Applying the power rule to the given equation, noting the constants in the first and second terms:

Then check to see if the critical point is a maximum, minimum, or an inflection point by taking the second derivative, using the power rule once again.

Because the second derivative is positive, the critical point is a minimum.

To find the point where the minimum occurs, plug back into the original equation and solve for .

Therefore, the minimum is

This problem requires the use of the product rule.

The product rule tells us that the derivative of,

is .

In this problem, is , and is .

Therefore,

, and = , so

, and factoring out gives us .

To solve, take the first derivative and evaluate at . Thus,

To find the derivative of this function, recall the product rule. You multiply the first expression by the derivative of the second and the add that to the product of derivative of the first expression by the second. Therefore, . Then, multiply and combine like terms so that your final answer is: .

The local minimum of a function can be found by finding the derivative and graphing it. The point in which the x axis is crossed from below gives the x position where the local minimum is found. Taking the derivative:

The graph of the derivative is shown below:

As shown by the graph, the local minimum is found at x = -4.

Using integrating factors:

Once we use this integrating factor, we can reduce the equation to:

Integrating both sides:

In the problem above, we can consider to be a composite function , where and .

According to the Chain Rule, .

Applying this rule to our composite function:

.

To find the slope of a curve at a certain point, you must first find the derivative of that curve.

The derivative of this curve is .

Then, plug in for into the derivative and you will get as the slope of at .