![]() ![]() "Well, what is f prime of "I don't know, negative three?" Well, you would evaluate X squared plus seven," and then they said, "All right, f of x, "f of x is equal to the fourth root of "x to the third plus four And so if someone were to tell you, if someone were to say, That's what we did here, times the derivative of the So I can take theĭerivative of the outside with respect to the inside, Something to the 1/4 power, basic exponent property, and then realize, okay, I have a composite function here. Realize is the fourth root is the same thing as taking Derivative of the outside, well, actually, the first thing to Recognize that this is an application of the chain rule. And you can manipulate this inĭifferent ways, if you like, but the key is to just 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. This up a little bit, so this would be equal to, this would be equal to. Multiply by u prime of x which is three x squared plus, three x squared plus eight x. Respect to x of a constant is just gonna be zero. With respect to x of seven, well, the derivative with Just one power, first power, so I can just write that as eight x, and then the derivative 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 Then I'm gonna multiply that times the derivative of the inside. I took the derivative of the outside with respect to the inside and And then I wanna multiply that, and this is the chain rule. So, this is going to be x to the third plus four x squared plus seven. U of x in here if I like, actually let me just do that. With respect to u of x, with respect to this I'm just gonna bring that 1/4 out front, so it's gonna be 1/4 times whatever I'm taking theĭerivative with respect to, to the 1/4 minus one power. Of that with respect to the inside, with respect to u of x. 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 Take the derivative, we would take the derivative of this, you could view it as the outer function with respect to u of xĪnd then multiply that times the derivative And then whatever we get for u of x, we raise that to the fourth power. What do we do first with our x? Well, we do all of this business, and we can call this u of x. I said a few seconds ago, we can view this as a composite function. The derivative of this? Well, we can view this, as The third plus four x squared plus seven to the 1/4 power, to the 1/4 power. Thing as the derivative with respect to x of x to And just realize that this fourth root is nothing but a fractional exponent. Make this fourth root a little bit more tractable for us. With a composite function, the chain rule should be front of mind. ![]() Of another expression?" And you'd be right. I have a composite function, "I'm taking the fourth root And at first you might say, "All right, "how do I take theĭerivative of the fourth root "of something, it looks like We can take the derivative with respect to x of the fourth root of x to the third power plusįour x squared plus seven.
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