# High School Math : Finding Derivatives

## Example Questions

### Example Question #53 : Derivatives

What is the second derivative of ?

Explanation:

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.

### Example Question #54 : Derivatives

What is the second derivative of ?

Explanation:

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

To find the first derivative of the problem, we can use the power rule. The power rule says to multiply the coefficient of the variable by the exponent of the variable and then lower the value of the exponent by one.

To make that work, we're going to treat  as , since anything to the zero power is one.

This means that  is the same as .

Now use the power rule:

Anything times zero is zero.

Now we repeat the process, but using .

Since anything times zero is zero,

### Example Question #55 : Derivatives

What is the second derivative of ?

Explanation:

To start we need to find the first derivative. For that, we need to use the chain rule:

Repeat the process!

Remember that :

### Example Question #56 : Derivatives

What is the second derivative of ?

Explanation:

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

To find the first derivative of , we can use the power rule.

The power rule states that we multiply each variable by its current exponent and then lower the exponent of each variable by one.

Since , we're going to treat  as .

Anything times zero is zero, so our final term , regardless of the power of the exponent.

Simplify what we have.

Our first derivative, then, is .

To find the second derivative, we repeat the process using  for our equation.

Simplify.

Remember that , which means our second derivative will be .

### Example Question #57 : Derivatives

If , what is ?

Explanation:

To find , or the second derivative of our function, we need to start by finding the first derivative.

To find the first derivative, we can use the power rule. The power rule states that we multiply each variable by its current exponent and then lower that exponent by one.

Simplify.

Anything to the zero power is one, so .

Therefore, .

Now we repeat the process, but we use .

Remember, anything times zero is zero.

### Example Question #1 : Using The Chain Rule

If  what is the slope of the line at .

Explanation:

The slope at any point on a line is also equal to the derivative. So first we want to find the derivative function of this function and then evaluate it at. So, to find the derivative we will need to use the chain rule. The chain rule says

so if we let  and  then

since  and

Therefore we evaluate at  and we get  or .

### Example Question #2 : Using The Chain Rule

What is the first derivative of ?

Explanation:

To solve for the first derivative, we're going to use the chain rule. The chain rule says that when taking the derivative of a nested function, your answer is the derivative of the outside times the derivative of the inside.

Mathematically, it would look like this:

Plug in our equations.

### Example Question #1 : Using The Chain Rule

Explanation:

For this problem we need to use the chain rule:

### Example Question #71 : Calculus I — Derivatives

Find the derivative of the following function:

Explanation:

Use -substitution so that .

Then the function  becomes .

By the chain rule, .

We calculate each term using the power rule:

Plug in :

### Example Question #72 : Calculus I — Derivatives

An ellipse is represented by the following equation:

What is the slope of the curve at the point (3,2)?

undefined

Explanation:

It would be difficult to differentiate this equation by isolating . Luckily, we don't have to.  Use to represent the derivative of  with respect to and follow the chain rule.

(Remember, is the derivative of  with respect to , although it usually doesn't get written out because it is equal to 1. We'll write it out this time so you can see how implicit differentiation works.)

Now we need to isolate by first putting all of these terms on the same side:

This is the equation for the derivative at any point on the curve. By substituting in (3, 2) from the original question, we can find the slope at that particular point: