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Differentiation calculus chain rule11/11/2023 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. Which represents the slope of the tangent line at the point (−1,−32).That y is equal to seven to the x squared minus x power. A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken.Įxample 1: Find f′( x) if f( x) = (3x 2 + 5x − 2) 8.Įxample 2: Find f′( x) if f( x) = tan (sec x).Įxample 5: Find the slope of the tangent line to a curve y = ( x 2 − 3) 5 at the point (−1, −32).īecause the slope of the tangent line to a curve is the derivative, you find that Here, three functions- m, n, and p-make up the composition function r hence, you have to consider the derivatives m′, n′, and p′ in differentiating r( x). If a composite function r( x) is defined as Note that because two functions, g and h, make up the composite function f, you have to consider the derivatives g′ and h′ in differentiating f( x). For example, if a composite function f( x) is defined as The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary.
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