# Precalculus : Find the Second Derivative of a Function

## Example Questions

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### Example Question #1 : Find The Second Derivative Of A Function

Find the second derivative of the following function:

Explanation:

In order to take any order derivative of a polynomial, all we need to know is how to apply the power rule to a simple term with an exponent:

The formula above tells us that to take the derivative of a term with coefficient  and exponent , we simply multiply the term by and subtract 1 from in the exponent. With this in mind, we'll take the first derivative of the given function, and then apply the power rule to each term once again to find the second derivative of the given function:

Now if we take the derivative of the first derivative, we'll get the second derivative of our function:

### Example Question #2 : Find The Second Derivative Of A Function

Find the second derivative of the function .

Explanation:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Taking the second derivative:

### Example Question #3 : Find The Second Derivative Of A Function

Find the second derivative of the function

Explanation:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Finally, we take the second derivative:

### Example Question #1 : Find The Second Derivative Of A Function

Find the second derivative of the function

Explanation:

To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient  and an exponent :

Applying this rule to each term in the function, we start by taking the first derivative:

Then, taking the second derivative of the function:

### Example Question #2 : Find The Second Derivative Of A Function

What is the second derivative of

with respect to

Explanation:

We first apply Power Rule.

First Derivative :

So result is

Anything to a power of  is

First Derivative is

Second Derivative :

Any derivative of a constant is

Second Derivative of  with respect to  is

### Example Question #3 : Find The Second Derivative Of A Function

Find the second derivative of

with respect to

Explanation:

Use Power Rule to take two derivatives of :

First Derivative:

So result is:

Now we take another derivative:

Second Derivative:

So our result is:

### Example Question #72 : Introductory Calculus

What is the second derivative of

with respect to .

Explanation:

Apply Power Rule twice.

First Derivative of :

So our result is

Second Derivative of :

So our result is

So the second derivative of  is

### Example Question #73 : Introductory Calculus

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 #74 : Introductory Calculus

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 #4 : 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:

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