Video: Finding the Sum of the Second Derivatives of Two Functions

Given that 𝑦 = (π‘₯ βˆ’ 7)(4π‘₯ + 7), and 𝑧 = π‘₯Β² + 5π‘₯ + 9, determine 𝑑²𝑦/𝑑π‘₯Β²+ 𝑑²𝑧/𝑑π‘₯Β².

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Video Transcript

Given that 𝑦 equals π‘₯ minus seven multiplied by four π‘₯ plus seven and 𝑧 equals π‘₯ squared plus five π‘₯ plus nine, determine 𝑑 squared 𝑦 over 𝑑π‘₯ squared plus 𝑑 squared 𝑧 over 𝑑π‘₯ squared.

Now, in order to actually solve this problem, what we’re gonna need to do is actually differentiate each of our functions and then differentiate them again to find the second-order derivative. So I’m gonna start with our first function, which is 𝑦 is equal to π‘₯ minus seven multiplied by four π‘₯ plus seven.

Well, the first thing we’re actually gonna have to do here is expand the parentheses. So the first term we’re gonna get is 𝑦 is equal to four π‘₯ squared cause we’ve got π‘₯ multiplied by four π‘₯, which gives us four π‘₯ squared. Then, we get plus seven π‘₯ because we have π‘₯ multiplied by positive seven minus 28π‘₯ cause we had negative seven multiplied by four π‘₯. And then, finally, we’ve got minus 49. So we’ve got four π‘₯ squared plus seven π‘₯ minus 28π‘₯ minus 49.

So what we can do next is actually simplify this by collecting like terms. So we’re gonna get 𝑦 is equal to four π‘₯ squared minus 21π‘₯ minus 49. Okay, fab, so this is out in a form, which is much easier to differentiate.

So what we’re gonna do first of all is actually find the first-order derivative. And when we actually differentiate four π‘₯ squared minus 21π‘₯ minus 49, we get eight π‘₯ minus 21. And just a quick recap of why we got that, we’re gonna look at the first term. Well, if we’ve got four multiplied by two and that’s because you have the coefficient multiplied by the exponent and then π‘₯ to the power of and then we got two minus one and that’s cause you subtract one from the exponent, so that leaves us with eight π‘₯.

So therefore, that’s how we actually differentiated each term. So we’ve got eight π‘₯ minus 21 as our first-order derivative. So therefore, what we want to do now is actually find the second-order derivative.

And the way we’re gonna do that is by finding 𝑑 squared 𝑦 over 𝑑π‘₯ squared. And to do that, we actually differentiate eight π‘₯ minus 21. And when we do that, we’re just gonna get the result as eight. And that’s because eight π‘₯ differentiates to just eight because again if you multiply the exponent by the coefficient, you’d have eight by one, which gives us eight. And then, you reduce the exponent of π‘₯ by one. So you go from π‘₯ to the power of one to π‘₯ to the power of zero, which would just give us one. So we’re just left with eight because negative 21 differentiates to zero.

So great! We’ve actually found 𝑑 squared 𝑦 over 𝑑π‘₯ squared for our first function. Now, let’s move on to our second function.

So looking at our second function, we’ve got 𝑧 is equal to π‘₯ squared plus five π‘₯ plus nine. So therefore, if we actually differentiate this, what we’re gonna get is 𝑑𝑧 𝑑π‘₯ is equal to two π‘₯ plus five. And that’s because again we use the same method that we had for the previous part. So if we differentiate the first term, we’d have the coefficient which is one multiplied by two, the exponent gives us two. And then, you reduce the exponent by one. So we just get two π‘₯.

Okay, great, so we found that. So again, what we need to do is actually find the second-order derivative. So in order to do that, so we’re gonna find 𝑑 squared 𝑧 over 𝑑π‘₯ squared. So we just differentiate two π‘₯ plus five. And when we do that, we’re just left with two.

So great, we’ve now actually found the second derivative of 𝑦 equals π‘₯ minus seven multiplied by four π‘₯ plus seven and the second derivative of 𝑧 equals π‘₯ squared plus five π‘₯ plus nine.

So now, let’s look back at the question what we need to do next. Well, we can see that actually we need to add our results together.

So therefore, what we can say is that given that 𝑦 equals π‘₯ minus seven multiplied by four π‘₯ plus seven and 𝑧 equals π‘₯ squared plus five π‘₯ plus nine, then 𝑑 squared 𝑦 over 𝑑π‘₯ squared plus 𝑑 squared 𝑧 over 𝑑π‘₯ squared is gonna be equal to two plus eight, which is equal to 10.

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