# High School Math : Derivatives

## Example Questions

### Example Question #11 : Finding Derivatives

Explanation:

To find the 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 exponent value 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.

### Example Question #12 : Finding Derivatives

What is the first derivative of ?

Explanation:

To find the 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 final solution, then, is .

### Example Question #21 : General Derivatives And Rules

If , what is ?

There is no  for this equation.

Explanation:

For this problem, 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, .

### Example Question #22 : General Derivatives And Rules

Find the derivative of the following function:

Explanation:

We use the power rule on each term of the function.

The first term

becomes

.

The second term

becomes

.

The final term, 7, is a constant, so its derivative is simply zero.

### Example Question #1 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

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

To take the derivative of this equation, we can 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.

Now we repeat the process, but using  as our expression.

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

Notice that  since anything times zero is zero.

### Example Question #2 : Finding Second Derivative Of A Function

What is the second derivative of ?

Undefined

Explanation:

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

To take the derivative of this equation, we can 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.

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

Notice that  since anything times zero is zero.

Simplify.

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

Now we repeat the process but use , or , as our expression.

As stated before, anything times zero is zero.

Therefore, .

### Example Question #3 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

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

### Example Question #4 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

To find the second derivative, we start by taking 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.

Now we repeat the process, but using  as our expression.

### Example Question #5 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

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

To find the first derivative, we can use the power rule. To do that, we lower the exponent on the variables by one and multiply by the original 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 repeat the process but using  as our expression.

Remember, anything to the zero power is one.

### Example Question #6 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

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

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.

Now we repeat the process, but we 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.