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Differential calculus chain rule12/18/2023 ![]() ![]() ![]() We can take a more formal look at the derivative of by setting up the limit that would give us the derivative at a specific value in the domain of. In addition, the change in forcing a change in suggests that the derivative of with respect to, where, is also part of the final derivative. First of all, a change in forcing a change in suggests that somehow the derivative of is involved. This chain reaction gives us hints as to what is involved in computing the derivative of. We can think of this event as a chain reaction: As changes, changes, which leads to a change in. Consequently, we want to know how changes as changes. We can think of the derivative of this function with respect to as the rate of change of relative to the change in. To put this rule into context, let’s take a look at an example. Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. And we're done, and we could distribute this natural log of four if we found that interesting.When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. The natural log of four times X squared plus X. And so we can just rewrite this as two X plus one over over over the natural log of four. And then we're gonna multiply that times U prime of X. So this is one over the natural log of four. Suppose a function y(x) y ( x ) is given implicitly, that is, y(x) y ( x ) satisfies some equation F(x,y)0 F ( x, y ) 0. More steps just hopefully so it's clearer what I'm doing here. And of course, that whole thing times U prime of X. Its hard to get, its hard to get too far in calculus without really grokking, really understanding the chain rule. This is just a review, this is the chain rule that you remember from, or hopefully remember, from differential calculus. Times, instead of putting an X there it would be times U of X. The derivative of f with respect to x, and thats going to give you the derivative of g with respect to x. So it's going to be one over the natural log of four. And you just do is you take the derivative of the green function with So, this is going to be equal to V prime of U X, U of X. If we want to know what V prime of U of X we would just replace wherever we see an X with a U of X. ![]() Well, what is V prime of U of X? We know what V prime of X is. This is going to be this is going to be the derivative of V with respect to U. And so we know from the chain rule the derivative Y with respect to X. And let me draw a little line here so that we don't get We have the whole expression that defines U of X. Remember, V is the logīase four of something. Information because Y this Y can be viewed as V of V of. You think of scaling the whole expression by one over the natural log of four. But we just scale it in the denominator with this natural log of four. Out of the change of base formulas that you might have seen. But since it's log base four and this comes straight If this was V of X, if V of X was just natural log of X, ourĭerivative would be one over X. So it's going to be one over one over log base four. Similar that if this was log base E, or natural log, except we're going to scale it. And then we've shown in other videos that V prime of X is, we're gonna be very The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function f g in terms of the derivatives of f and g. We can say V of well if we said V of X this would be log base four of X. In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. ![]() And we can say the log base four of this stuff well we could call that a function V. Derivative with respect to X of X is one. So that's gonna be I'm just gonna use the power rule here so two X plus one I brought that two out front and decremented the exponent. And it's gonna be useful later on to know what U prime of X is. Given a R and functions f and g such that g is. So we could say we could say this thing in blue that's U of X. f(x2), were ready to explore one of the power tools of differential calculus. We're taking the logīase four, not just of X, but we're taking that What is the derivative of Y with respect to X going to be equal to? Now you might recognize immediately that this is a composite function. Let's say that Y is equal to log base four of X squared plus X. ![]()
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