Worked example: Derivative of ∜(x³+4x²+7) using the chain rule

- [Voiceover] Let's see if we can take the derivative with respect to x of the fourth root of x to the third power plus four x squared plus seven. And at first you might say, "All right, "how do I take the derivative of the fourth root "of something, it looks like I have a composite function, "I'm taking the fourth root of another expression?" And you'd be right. And if you're dealing with a composite function, the chain rule should be front of mind. But first, let's just make this fourth root a little bit more tractable for us. And just realize that this fourth root is nothing but a fractional exponent. So this is the same thing as the derivative with respect to x of x to the third plus four x squared plus seven to the 1/4 power, to the 1/4 power. Now, how do we take the derivative of this? Well, we can view this, as I said a few seconds ago, we can view this as a composite function. What do we do first with our x? Well, we do all of this business, and we can call this u of x. And then whatever we get for u of x, we raise that to the fourth power. So the way that we would take the derivative, we would take the derivative of this, you could view it as the outer function with respect to u of x and then multiply that times the derivative of u with respect to x. So let's do that. So what this is going to be, this is going to be equal to, so we're gonna take our outside function, which I'm highlighting in green now, so, or I take something to the 1/4, I'm gonna take the derivative of that with respect to the inside, with respect to u of x. Well, I'm just gonna use the power rule here. I'm just gonna bring that 1/4 out front, so it's gonna be 1/4 times whatever I'm taking the derivative with respect to, to the 1/4 minus one power. Look, all I did was use the power rule here. I didn't have an x here. Now I'm taking the derivative with respect to u of x, with respect to this polynomial expression here. So I could just throw the u of x in here if I like, actually let me just do that. So, this is going to be x to the third plus four x squared plus seven. And then I wanna multiply that, and this is the chain rule. I took the derivative of the outside with respect to the inside and then I'm gonna multiply that times the derivative of the inside. So what's the derivative of u of x? U prime of x, let's see we just gonna use the power rule a bunch of times, it's gonna be three x squared plus two times four is eight x to the two minus one is just one power, first power, so I can just write that as eight x, and then the derivative with respect to x of seven, well, the derivative with respect to x of a constant is just gonna be zero. So that's u prime of x. So then I'm just gonna multiply by u prime of x which is three x squared plus, three x squared plus eight x. And so, well, I can clean this up a little bit, so this would be equal to, this would be equal to. Actually let me just rewrite that exponent there. So this 1/4 minus one, I can rewrite it, 1/4 minus one is negative 3/4, negative 3/4, negative 3/4 power. And you can manipulate this in different ways, if you like, but the key is to just recognize that this is an application of the chain rule. Derivative of the outside, well, actually, the first thing to realize is the fourth root is the same thing as taking something to the 1/4 power, basic exponent property, and then realize, okay, I have a composite function here. So I can take the derivative of the outside with respect to the inside, that's what we did here, times the derivative of the inside with respect to x. And so if someone were to tell you, if someone were to say, "All right, f of x, "f of x is equal to the fourth root of "x to the third plus four x squared plus seven," and then they said, "Well, what is f prime of "I don't know, negative three?" Well, you would evaluate this at negative three. So let me just do that. So it's 1/4 times, see, you have negative 27, I hope this works out reasonably well, plus 36, plus 36, plus 7 to the negative 3/4, what does this result to? This is going to be equal to, this right over here is 16, right? Negative 27 plus seven is negative 20 plus 36, so this is 16. I think this is going to work out nicely. And then times three times negative, so three times nine, which is 27 minus 24. So this is going to be right over here, that is going to be three. Now, what is 16 to the negative 3/4? So this is 1/4 times, so 16 times to the 1/4 is two and then you raise that to the, let me, actually I don't want to skip steps here. At this point we're dealing with algebra, or maybe even pre-algebra. So this is going to be times, times, 16 to the 1/4, and then we're gonna raise that to the negative three times that three out front. So we could put that three there. 16 to the 1/4 is two, two to the third is eight, so two to the negative third power is 1/8, so that is 1/8. So we have 3/4 times 1/8 which is equal to three over 32, 3/32. So that would be the slope of the tangent line of the graph y is equal to f of x when x is equal to negative three.