Finding Second Derivative of a Function - Math

Take the derivative of , then take the derivative of .

Take the derivative of , then take the derivative of .

Rewrite:

Take the derivative of , then take the derivative of .

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

Repeat the process!

Remember that :

Take the derivative of , then take the derivative of .

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.

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.

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 .

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.

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 .

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.

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.

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, .

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.

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.

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.

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.

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.

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 .