So lowercase-F-prime of g of x times the derivative Restate the chain rule, the derivative of capital-F is going to be theĭerivative of lowercase-f, the outside function with Viewed as the composition of other functions that theĬhain rule will apply here. Recognize that if I have a function that can be It's lowercase-f of g of x, and they want us toĮvaluate f-prime of four. Let function capital-F beĭefined as the composition of f and g. X equals negative two, x equals four, they gave us the values of f, g, f-prime, and g-prime. That times u prime of x, and once again this is just an application of the chain rule.Table lists the values of functions f and g and of their derivatives,į-prime and g-prime for the x values negative two and four. Gonna be natural log of seven times seven to the u of x power, we take that and multiply Way that we would've treated an x right over here, so it's It in different ways, but the main thing to realize is, look, we're just gonna take theĭerivative of the seven to the this to the u of x You could, we could try to simplify this or I guess re-express Multiply this times two x, two x minus one, so there you have it, that is the derivative Is just two x minus one, so we're going to So, u prime of x, well, that's going to be two x to the first which That times the derivative of u with respect to x. The x squared minus x power, x squared, x squared minus x power, and then we want to multiply Here, this is going to be natural log of seven times seven to the, instead of saying seven to the x power, remember we're taking v prime of u of x, so it's going to be seven to So, if we are taking v prime of u of x, then notice instead of an x everywhere, we're going to have a u of x everywhere. Going to be the natural log of seven times seven to the x power. Take derivatives exponentials of bases other than e, this To, and we've proved this in other videos where we Let me actually write it right over here, if v of x isĮqual to seven to the x power v prime of x would be equal Of v with respect to u? What is v prime of u of x? Well, we know, we know, Of u with respect to x, derivative of u with respect to x, and so either way we canĪpply that right over here. Of u with respect to x, so that's one way you could do it, or you could say that this is equal to, this is equal to theĭerivative, the derivative of v with respect to x, sorry,ĭerivative of v with respect to u, d v d u times the derivative U, so v prime of u of x times the derivative Written as the derivative of v with respect to Y with respect to x, and you'll see different notations here, sometimes you'll see it The chain rule tells us that the derivative of To x squared minus x, then what we have right over here, y, y is equal to seven to something, so it's equal to v of,Īnd it's not just v of x, it's v of u of x, instead of an x here you have the whole function u of x, x squared minus x. If you had a function v of x, which is equal to seven to the xth power, and you had another function u of x, u of x which is equal Or it could be viewed as a composite function. Well, based on how this hasīeen color-coded ahead of time, you might immediately recognize that this is a composite function, Of y, derivative of y, with respect to x? And like always, pause this video and see if you can figure it out. That y is equal to seven to the x squared minus x power.
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