Derivatives - AP Calculus AB

The derivative of is . (Memorization)

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

Plugging in , we get

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

Now we plug in

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 = .

The answer is .

The second derivative is just the derivative of the first derivative. So first we find the first derivative of . Remember the derivative of is , and the derivative for is .

Then to get the second derivative, we just derive this function again. So

Take the derivative of , then take the derivative of .

Take the derivative of , then take the derivative of .

Rewrite:

Take the derivative of , then take the derivative of .

Take the derivative of , then take the derivative of .

To get the second derivative, first we need to find the first derivative.

To do that, 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.

Now we do the same process again, but using as our expression:

Notice that , as anything times zero will be zero.

Anything to the zero power is one.

To get the second derivative, first we need to find the first derivative.

To do that, 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.

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

That means this problem will look like this:

Notice that as anything times zero will be zero.

Remember, anything to the zero power is one.

Now to get the second derivative we repeat those steps, but instead of using , we use .

Notice that as anything times zero will be zero.

To get the second derivative, first we need to find the first derivative.

To do that, 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.

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

Notice that , as anything times zero is zero.

Now we repeat the process using as the expression.

Just like before, we're going to treat as .

First, find the double derivative of the function given.

You should get .

Next, plug in x=1 to get .

The double derivative is a positive number, which means the concavity is Concave Up.

The question is asking us for the second derivative of the equation. First, we need to find the first derivative.

To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

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

Notice that since anything times zero is zero.

Now we do the exact same process but using as our expression.

As stated earlier, anything to the zero power is one.

To find the second derivative, first we need to find the first derivative.

To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

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

Notice that since anything times zero is zero.

That leaves us with .

Simplify.

As stated earlier, anything to the zero power is one, leaving us with:

Now we can repeat the process using or as our equation.

As pointed out before, anything times zero is zero, meaning that .

To find the second derivative, first we need to find the first derivative.

To do that, we can use the power rule. To use the power rule, we lower the exponent on the variable and multiply by that exponent.

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

Notice that since anything times zero is zero.

Just like it was mentioned earlier, anything to the zero power is one.

Now we repeat the process using as our expression.

Like before, anything times zero is zero.

Anything to the zero power is one.

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.

To find out if the function is increasing or decreasing, we need to look at the first derivative.

To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.

Anything to the zero power is one.

Now we plug in our given value and find out if the result is positive or negative. If it is positive, the function is increasing. If it is negative, the function is decreasing.

Therefore, the function is decreasing.

To find out if it is concave or convex, look at the second derivative. If the result is positive, it is convex. If it is negative, then it is concave.

To find the second derivative, we repeat the process using as our expression.

We're going to treat as .

Notice that since anything times zero is zero.

As stated before, anything to the zero power is one.

Since we get a positive constant, it doesn't matter where we look on the graph, as our second derivative will always be positive. That means that this graph is going to be convex at our given point.

Therefore, the function is decreasing and convex at our given point.

First, let's find out if the graph is increasing or decreasing. For that, we need the first derivative.

To find the first derivative, we can use the power rule. We lower the exponent on all the variables by one and multiply by the original variable.

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

Notice that since anything times zero is zero.

Plug in our given point for . If the result is positive, the function is increasing. If the result is negative, the function is decreasing.

Our result is negative, therefore the function is decreasing.

To find the concavity, look at the second derivative. If the function is positive at our given point, it is concave. If the function is negative, it is convex.

To find the second derivative we repeat the process, but using as our expression.

As you can see, our second derivative is a constant. It doesn't matter what point we plug in for ; our output will always be negative. Therefore our graph will always be convex.

Combine our two pieces of information to see that at the given point, the graph is decreasing and convex.

is increasing on those intervals at which .

We need to find the values of for which . To that end, we first solve the equation:

These are the boundary points, so the intervals we need to check are:

, , and

We check each interval by substituting an arbitrary value from each for .

Choose

increases on this interval.

Choose

decreases on this interval.

Choose

increases on this interval.

The answer is that increases on