Find the Second Derivative of a Function - Pre-Calculus

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:

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:

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:

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:

Use Power Rule to take two derivatives of :

First Derivative:

So result is:

Now we take another derivative:

Second Derivative:

So our result is:

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

Apply Power Rule twice.

First Derivative of :

So our result is

Second Derivative of :

So our result is

So the second derivative of is

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:

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:

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:

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:

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

Therefore, taking each term of :

Then, taking the derivative again:

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

Taking the first derivative of :

Then, taking the second derivative of :

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

Taking the first derivative of :

Then, taking the second derivative:

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.

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.

We first need to find the first derivative of . Remember that according to the derivatives of trigonometric functions, the derviative of cosine is negative sine and the derivative of sine is cosine.

Applying these rules we are able to find the first derivative.

Now to find the second derivative we take the derivative of the first derivative.

To find the second derivative of we will need to take the derivative of the first derivative.

To find the first derivative we will use the rule for natural logs which states,

Applying this rule to our function we get the following.

Now that we have the first derivative we will take the derivative of it to get the second derivative. In order to do so we will need to use the quotient rule which states,

Applying this rule we get the following.

To find the second derivative of this function we will need to use the following rules.

Product Rule,

.

Chain Rule,

.

Applying these rules we can find both the first derivative and then the second derivative.

The derivative of this function, known as (read as f prime of x), can be found by using the Power rule of derivatives on each term in the function. Power rule: . If there is a value in front of x we multiply it by the "n" we carried over. For example taking the derivative of . To find the second derivative, simply repeat the process. is read as "f double prime of x" or "the second derivative of f(x)".