Derivative Functions - Example 2

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity, and the derivative of velocity with respect to time is acceleration. The concept of a derivative has been generalized to other contexts than just functions of a single real variable. For instance, the derivative of a function of several real variables is a function of several real variables, and the derivative of a function from Rn to Rm is a function from Rn to Rm.

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Video Transcript

All right. So we want to find the second derivative function of this function ethics. Okay, so first things first. How do you find the second driven? The second derivative is the derivative of the first trip. Okay, so let's just find the first derivative first. So the first derivative is gonna be the limit. His H goes to zero of X plus h cubed minus three X plus H squared minus 24 X plus H minus seven. Okay, so that's FX plus age minus X cubed minus three X squared minus 24 x plus. So oh, over H. And of course, if you try to plug in H equals zero, you're going to get an indeterminant form. Unfortunately, we have to expand and hope a lot of things canceled in the new greater. So let's expand the numerator. And in fact, let's just expand the numerator and then right with the limit equals so expanding the numerator. So we have X cubed from this term X cubed. I'm just gonna use kind of the binomial theorem three x squared H plus three x h squared plus H cube. Okay, that's from this first term and then minus three X squared minus six X h minus three h squared because that's coming from this term. So I just multiply this finer mealtime itself multiplied by negative three and then minus 24 X minus 24 age minus seven. But the minus seven we're gonna cancel you see that right away? Okay, so we just have minus execute plus three x squared plus 24 X. Okay, so that's what the numerator is. It's a mess. It's very easy to make mistakes. So probably it's a good idea to write this out big and clear what I'm doing here and let's see what Kansas X Cubed cancels minus X cubed. We have minus three x squared plus three x squared. Then we have minus 24 X plus 24 tracks. So all in all, what is the numerator become? So the derivative is the limit. It is a joke. Goes to zero of the numerator, which is three x squared H plus three x h squared plus H cube minus six X H minus three h squared minus 24 h Oliver age. Okay, And let's cancel one factor at age. So as we kind of expected everything is gonna have a factor of age. But now if we take ht zero well, three x squared. That doesn't happen. Age. So that's left. This still has a factor of age, so that goes to 03 x h h goes to zero here, H goes to zero. Here, this is just minus six X. So that's left this three h has a factor of h. So that goes to zero his age close to zero. And then we have minus 24. Okay, so that's the derivative. Easy enough supreme for the second derivative. We just take the derivative of the first trip. So it's still limit his h goes to zero. But now, instead of plugging an F of X plus age ffx, we're gonna plug in F prime of X plus H minus f prime of X. So this is gonna be three times X plus H squared minus six has expose age minus 24 minus F prime attacks, which is three X squared, minus six x minus 24. And of course, all over H. So you see the important thing to notice here. Is it the second derivative? It's just the derivative of the first derivative. So we're doing the same thing we did upon the derivative except using that crime instead of that. So now if we simplify this out again, a lot of things should cancel. We have three x squared plus six x h plus three H Square minus six X minus six age minus 24 minus three. Expert distribute plus six x plus 24 all over age. So let's do it. Cancels three X squared minus three X squared minus six X plus six x minus 24 plus 24 We're left with six x h plus three h squared minus six age. We can cancel one factor of H and all we're left with is six x minus six plus three h. But his H goes to zero in the H goes to zero. So all we're left with it's six x minus six, and Matt is the second derivative of that function.