Find the third derivative of the given function. y = 3 x 2 + 9 x + 1 y=3x^2+9x+1 y = 3 x 2 + 9 x + 1

In this problem, we're asked to find the 3rd derivative of the given function, which here is y equals 3x squared plus 9x plus 1. Now in order to find the 3rd derivative, we need to take the derivative, you guessed it, three times. So let's get started in finding that first derivative so that we can get to the 3rd. Now to find our first derivative here, I'm gonna denote that as y prime. I could also choose to denote that as dydx. Either of those notations are correct, but I'm going to use primes here. So y prime, my first derivative here, looking at that first term, I'm going to use the power rule, pulling that 2 out to the front, multiplying by it, and decreasing that power by 1. This will give me 6x to the power of 2 minus 1, which is just one, so I don't need to worry about writing that exponent down. Then for this next term, I have 9 x. Now remember that if I have a constant times x, my derivative will always just end up being that constant, so this is going to give me plus 9. My final term is just a constant by itself. It's 1, and I know that the derivative of any constant is just 0, so I don't need to worry about adding anything else here. So this first derivative, 6x+9. Now let's move on to our second derivative because we are not done yet. We need to get all the way up to our 3rd derivative here. Now for my second derivative, y double prime, here I have 6x+9. Now looking at this, I have this 6x here. I know that if I have a constant times x, my derivative is just that constant. So I get 6 here. Then that second term is just a constant. So I know that derivative is 0. I don't need to add anything else here for my second derivative, which is just 6. Now finally, we want to find our 3rd derivative. This will be our final answer. And since I'm just taking the derivative of a constant here, I know that the derivative of any constant, no matter what it is, is 0. So I am all done here. My third derivative is going to be 0. Now if you were curious, if we kept going and finding more higher order derivatives, our 4th, 5th, 6th, 1 hundredth derivative, once you get to 0, all of your higher order derivatives beyond that are also 0. So at this point, if they ask you to find the 100th derivative, you would already know that it's just 0. Feel free to let me know if you have any questions here, and let's get some more practice.

3. Techniques of Differentiation

Higher Order Derivatives

Business Calculus

3. Techniques of Differentiation

Higher Order Derivatives

Struggling with Business Calculus?

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Multiple Choice

Find the third derivative of the given function.
$y=3x^2+9x+1$

$0$

$6$

$x$

$3x^2$

Verified step by step guidance

Start by identifying the given function: $ y = 3x^2 + 9x + 1 $. The goal is to find the third derivative, which means we will differentiate the function three times.

First, compute the first derivative $ y' $ by applying the power rule $ \frac{d}{dx}[x^n] = n \cdot x^{n-1} $: $ y' = \frac{d}{dx}[3x^2] + \frac{d}{dx}[9x] + \frac{d}{dx}[1] $.

Simplify the first derivative: $ y' = 6x + 9 $, since the derivative of $ 3x^2 $ is $ 6x $, the derivative of $ 9x $ is $ 9 $, and the derivative of a constant $ 1 $ is $ 0 $.

Next, compute the second derivative $ y'' $ by differentiating $ y' $: $ y'' = \frac{d}{dx}[6x] + \frac{d}{dx}[9] $.

Simplify the second derivative: $ y'' = 6 $, since the derivative of $ 6x $ is $ 6 $ and the derivative of a constant $ 9 $ is $ 0 $. Finally, compute the third derivative $ y''' $: $ y''' = \frac{d}{dx}[6] $, which simplifies to $ 0 $, as the derivative of a constant is always $ 0 $.