# Precalculus : Derivatives

## Example Questions

### Example Question #5 : Maximum And Minimum Problems

Find the coordinates of the relative maximum point of the function

.

Explanation:

Let's find the first derivative to locate the relative maxima and minima.

Now we set it equal to zero to find the x values of these critical points.

So the equation is 0 where x is -2, 0, or 5. Now let's find the second derivative so that we know which of these locations are maxima and which are minima.

So the function has a relative maximum at x=-5.

So the function has a relative minimum at x=0.

So the function has a relative maximum at x=2.

Thus there is only one relative minimum in this function, and it occurs at x=0. We need to plug this into the original function to find the y-coordinate of the point.

So our point is (0,8).

### Example Question #6 : Maximum And Minimum Problems

Find the -coordinates of the possible locations of the relative maxima and minimia of

.

There are no critical points.

Explanation:

We need to find out where the first derivative is equal to zero to find the locations of the possible maxima and minima.

or .

### Example Question #7 : Maximum And Minimum Problems

Without solving the problem, determine whether the function  will have relative maxima or minima and how many of each.

One minimum and one maximum

One relative maximum

No critical points

One relative minimum

One relative minimum

Explanation:

We see that the function is a quadratic, so its graph will be a parabola. Thus, we know it will have only one relative max or min. Since the leading coefficient of  is a positive 6, we also know that the parabola opens upward. Hence, the relative extremum is a minimum.

### Example Question #1 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

Explanation:

In order to take any order derivative of a polynomial, all we need to know is how to apply the power rule to a simple term with an exponent:

The formula above tells us that to take the derivative of a term with coefficient  and exponent , we simply multiply the term by and subtract 1 from in the exponent. With this in mind, we'll take the first derivative of the given function, and then apply the power rule to each term once again to find the second derivative of the given function:

Now if we take the derivative of the first derivative, we'll get the second derivative of our function:

### Example Question #2 : Find The Second Derivative Of A Function

Find the second derivative of the function .

Explanation:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Taking the second derivative:

### Example Question #3 : Find The Second Derivative Of A Function

Find the second derivative of the function

Explanation:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Finally, we take the second derivative:

### Example Question #4 : Find The Second Derivative Of A Function

Find the second derivative of the function

Explanation:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Then, taking the second derivative of the function:

### Example Question #5 : Find The Second Derivative Of A Function

What is the second derivative of

with respect to

Explanation:

We first apply Power Rule.

First Derivative :

So result is

Anything to a power of  is

First Derivative is

Second Derivative :

Any derivative of a constant is

Second Derivative of  with respect to  is

### Example Question #6 : Find The Second Derivative Of A Function

Find the second derivative of

with respect to

Explanation:

Use Power Rule to take two derivatives of :

First Derivative:

So result is:

Now we take another derivative:

Second Derivative:

So our result is:

### Example Question #7 : Find The Second Derivative Of A Function

What is the second derivative of

with respect to .

Explanation:

Apply Power Rule twice.

First Derivative of :

So our result is

Second Derivative of :

So our result is

So the second derivative of  is