# High School Math : Derivatives

## Example Questions

### Example Question #4 : Finding Regions Of Concavity And Convexity

When , what is the concavity of the graph of ?

Increasing, convex

Decreasing, convex

There is insufficient data to solve.

Decreasing, concave

Increasing, concave

Increasing, convex

Explanation:

To find the concavity, we need to look at the first and second derivatives at the given point.

To take the first derivative of this equation, use the power rule. The power rule says that we lower the exponent of each variable by one and multiply that number by the original exponent:

Simplify:

Remember that anything to the zero power is equal to one.

The first derivative tells us if the function is increasing or decreasing. Plug in the given point, , to see if the result is positive (i.e. increasing) or negative (i.e. decreasing).

Therefore the function is increasing.

To find out if the function is convex, we need to look at the second derivative evaluated at the same point, , and check if it is positive or negative.

We're going to treat  as  since anything to the zero power is equal to one.

Notice that  since anything times zero is zero.

Plug in our given value:

Since the second derivative is positive, the function is convex.

Therefore, we are looking at a graph that is both increasing and convex at our given point.

### Example Question #5 : Finding Regions Of Concavity And Convexity

At the point where , is  increasing or decreasing, and is it concave up or down?

Increasing, concave up

There is no concavity at that point.

Decreasing, concave down

Increasing, concave down

Decreasing, concave up

Increasing, concave up

Explanation:

To find if the equation is increasing or decreasing, we need to look at the first derivative. If our result is positive at , then the function is increasing. If it is negative, then the function is decreasing.

To find the first derivative for this problem, we can use the power rule. The power rule states that we lower the exponent of each of the variables by one and multiply by that original exponent.

Remember that anything to the zero power is one.

Plug in our given value.

Is it positive? Yes. Then it is increasing.

To find the concavity, we need to look at the second derivative. If it is positive, then the function is concave up. If it is negative, then the function is concave down.

Repeat the process we used for the first derivative, but use  as our expression.

For this problem, we're going to say that  since, as stated before, anything to the zero power is one.

Notice that  as anything times zero is zero.

As you can see, there is no place for a variable here. It doesn't matter what point we look at, the answer will always be positive. Therefore this graph is always concave up.

This means that at our given point, the graph is increasing and concave up.

### Example Question #1 : Finding Derivative At A Point

Find  if the function  is given by

Explanation:

To find the derivative at , we first take the derivative of . By the derivative rule for logarithms,

Plugging in , we get

### Example Question #2 : Finding Derivative At A Point

Find the derivative of the following function at the point .

Explanation:

Use the power rule on each term of the polynomial to get the derivative,

Now we plug in

### Example Question #3 : Finding Derivative At A Point

Let . What is ?

Explanation:

We need to find the first derivative of f(x). This will require us to apply both the Product and Chain Rules. When we apply the Product Rule, we obtain:

In order to find the derivative of , we will need to employ the Chain Rule.

We can factor out a 2x to make this a little nicer to look at.

Now we must evaluate the derivative when x = .

### Example Question #1 : General Derivatives And Rules

What is the first derivative of ?

Explanation:

To find the first derivative for this problem, we can use the power rule. The power rule states that we lower the exponent of each of the variables by one and multiply by that original exponent.

Remember that anything to the zero power is one.

### Example Question #2 : General Derivatives And Rules

Explanation:

This problem is best solved by using the power rule. For each variable, multiply by the exponent and reduce the exponent by one:

Treat as since anything to the zero power is one.

Remember, anything times zero is zero.

### Example Question #3 : General Derivatives And Rules

Give the average rate of change of the function  on the interval  .

Explanation:

The average rate of change of  on interval  is

Substitute:

### Example Question #1 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?

Explanation:

To solve this problem, we can use the power rule. That means we lower the exponent of the variable by one and multiply the variable by that original exponent.

Remember that anything to the zero power is one.

### Example Question #2 : Derivative Defined As The Limit Of The Difference Quotient

What is the derivative of ?