# Precalculus : Derivatives

## Example Questions

### Example Question #8 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

Explanation:

To find the second derivative of any function, we find the first derivative, and then just take the derivative again. If we take the first derivative, we apply the power rule and see that the exponent of x for the first term will drop to 0, which means it becomes a 1, leaving us only with the coefficient 127. The second term is just a constant, so its derivative is 0:

Now we can see that our first derivative is just a constant, so when we take the derivative again to find the second derivative we will end up with 0:

### Example Question #9 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

Explanation:

To find the second derivative of any function, we start by finding the first derivative. Looking at our function, we'll apply the power rule by bringing down each exponent and multiplying it by the coefficient of its term, then we'll subtract 1 from the new exponent:

Notice that the constant at the end drops off because the derivative of a constant is 0. Now we simply take the derivative one more time to find the second derivative:

### Example Question #10 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

Explanation:

To find the second derivative of any function, we start by finding the first derivative. We do this by applying the power rule to each term, multiplying each term by the value of its exponent and then subtracting 1 from the exponent to give its new value:

Now we simply take the derivative of the first derivative to find the second derivative:

### Example Question #11 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

Explanation:

We could multiply out our function and then find the second derivative, or we could apply the chain rule to find the first derivative and then apply the product rule to find the second derivative. Let's try the second method:

Now that we have our first derivative from the chain rule, we can find the second derivative using the product rule:

Which we can then multiply out and simplify the arrive at the following answer:

### Example Question #12 : Find The Second Derivative Of A Function

Find the second derivative of .

Explanation:

For any function , the first derivative   and the second derivative is

Therefore, taking each term of :

Then, taking the derivative again:

### Example Question #13 : Find The Second Derivative Of A Function

Find the second derivative of the function

Explanation:

For any function , the first derivative   and the second derivative is .

Taking the first derivative of :

Then, taking the second derivative:

### Example Question #14 : Find The Second Derivative Of A Function

Find the second derivative of the function

Explanation:

For any function , the first derivative   and the second derivative is .

Taking the first derivative of :

Then, taking the second derivative of :

### Example Question #15 : Find The Second Derivative Of A Function

Find the second dervative of the following equation.

Explanation:

To find the second derivative, simply differentiate the equation twice.

To differentiate this problem we will need to use the power rule which states,

.

Applying the power rule to each term twice, we are able to find the second derivative.

### Example Question #16 : Find The Second Derivative Of A Function

Find the second derivative of  with respect to  when

.

Explanation:

For this problem we will need to use the power rule on each term.

The power rule is,

Applying the power rule to our function we get the following derivative.

### Example Question #17 : Find The Second Derivative Of A Function

Find the second derivative of .