Here, we have figured out, is one over the square root of x, and this thing, u prime Of x, so v prime of u of x is going to be equal to, is going to be equal to one over u of x, one over u of x, which is equal to, which is equal to one over, u of x is just the square root of x. That a little bit neater, we replace that with a u Wherever we see the x, we replace it, let me write And so we now know what u prime of x is, we know what v prime of x is,īut what is v prime of u of x? Well v prime of u of x, The natural log of x is one over x, we show What is v prime of x? Well the derivative of Out of that exponent, so that's 1/2 minus one It becomes 1/2 x to the, and then take off one Of x is just the same thing as x to 1/2 power, so weĬan use the power rule, bring the 1/2 out from so Here, is going to be equal to, well remember, square root The derivative of u of x and v of x, u prime of x
#Khan academy ap calculus chain rule how to
So how do we evaluate these things? Well, we know how to take Of this inside function with respect to x. Us that f prime of x is going to be equal to the derivative of, you can view it as the outside function, with respect to this inside function, so it's going to be v prime of u of x, v prime of u of x, times the derivative Which tells you that, okay, if I'm trying toįind the derivative here, the chain rule is going to be
And as you can see here, f of x, and I color-coded ahead of time, is equal to, f of x is equal to, the natural log of the square root of x. That's the natural log, that is just the natural log of x. V with x as an input, we would just say well Is taking the natural log, the input happens to be square root of x, so it outputs the natural Of f, or in the case of how I just diagrammed it, v And then you take that output, and input it into anotherįunction that we could call v, and what does v do? Well it take the natural log Of whatever its input is, so u of x is equal to So you could have a function u here, which takes the square root You're inputting into one function then taking that output and That is f of x, which is essentially, a composition of two functions. X as this entire set, or this entire, I guess you could say, this combination ofįunctions right over there. And then what do you produce? Well you produce the natural I'm making these little squares to show what you do with the input. That takes the natural log of whatever is inputted in.
So then you take the natural log of that, so you could view that as inputting it into another function Square root of the input to produce the square root of x, and then what do you do? You take the square root and then you take the natural log of that. Some x, you input it, the first thing that you do,
X into our function f, what's the first thing that you do? Well, you take the square root of it. Out, what's going on here? Well if you input an Recognize that f can actually be viewed as a composition And what we wanna do in this video is find the derivative of f. So we have here f of x being equal to the natural
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