### All High School Math Resources

## Example Questions

### Example Question #51 : Derivatives

If , what is ?

**Possible Answers:**

**Correct answer:**

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.

### Example Question #52 : Derivatives

What is the second derivative of ?

**Possible Answers:**

Undefined

**Correct answer:**

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 .

### Example Question #53 : Derivatives

What is the second derivative of ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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 .

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

For this problem we need to use the chain rule:

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