High School Math : Derivatives

Example Questions

Example Question #7 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

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

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.

From here, we can use our normal power rule to find the second derivative.

Anything times zero is zero.

Anything to the zero power is one.

Example Question #8 : Finding Second Derivative Of A Function

What is the second derivative of ?

Explanation:

To find the second derivative, we need to find the first derivative first. To find the first derivative, we can use the power rule.

For each variable, multiply by the exponent and reduce the exponent by one:

Treat as since anything to the zero power is one.

Remember, anything times zero is zero.

Now follow the same process but for .

Therefore the second derivative will be the line .

Example Question #9 : Finding Second Derivative Of A Function

Let .

Find the second derivative of .

Explanation:

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

Example Question #10 : Finding Second Derivative Of A Function

Define .

What is ?

Explanation:

Take the derivative  of , then take the derivative of .

Example Question #31 : General Derivatives And Rules

Define .

What is  ?

Explanation:

Take the derivative  of , then take the derivative of .

Example Question #32 : General Derivatives And Rules

Define .

What is ?

Explanation:

Take the derivative  of , then take the derivative of .

Example Question #33 : General Derivatives And Rules

Define .

What is ?

Explanation:

Rewrite:

Take the derivative  of , then take the derivative of .

Example Question #34 : General Derivatives And Rules

What is the second derivative of ?

Explanation:

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.

Example Question #35 : General Derivatives And Rules

What is the second derivative of ?

Undefined

Explanation:

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.

Example Question #36 : General Derivatives And Rules

What is the second derivative of ?

Explanation:

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 .