![]() This is called the Generalized Power Rule.Įxample 62: Using the Chain Rule to find a tangent line ![]() Three times the two X which is going to be six X, so I've covered those so far times sin squared of X squared, times sin squared of X squared, times cosine of X squared.\cdot g^\prime (x)\). Of a mini drum roll here, this shouldn't take us too long, DY/DX, I'll multiply the Wanted to write the DY/DX, let me get a little bit That, we just use the power rule, that's going to be two X. Now we just have toįigure out the derivative with respect to X of X squared and we've seen that many times before. Something is our X squared and of course, we haveĪll of this out front which is the three times sin of X squared, I could write The derivative of this is gonna be the sin of something with respect to something, so that is cosine of that something times the derivative with respect to X of the something. So, I'm going to take the derivative, it's sin of something, so this is going to be, To now take the derivative of sin of X squared. This is just a matter of the first part of the expression is just a matter ofĪlgebraic simplification but the second part we need The orange parentheses and these orange brackets right over here. Of these orange parentheses I would put it inside of Times that something squared times the derivative with respect to X of that something, in this case, the something is sin, let me write that in the blue color, it is sin of X squared. Something to the third power with respect to that something. So, if we apply the chain rule it's gonna be theĭerivative of the outside with respect to the inside or the something to the third power, the derivative of the And so, one way to tackle this is to apply the chain rule. Outside of this expression we have some business in here that's being raised to the third power. This isn't a straightforwardĮxpression here but you might notice that I have something being raised to the third power, in fact, if we look at the Of this with respect to X? What is DY/DX which weĬould also write as Y prime? Well, there's a couple of ![]() Squared to the third power, which of course we could also write as sin of X squared to the third power and what we're curious about is what is the derivative I've spent a lot of time on this and now I think it is starting to make sense. And finally multiplies the result of the first chain rule application to the result of the second chain rule application.Įarlier in the class, wasn't there the distinction between outside and inside of an expression? Here we apply the chain rule to the outside first (the cube function), and secondly we apply it to the inside function, the sin(x^2). He then goes on to apply the chain rule a second time to what is inside the parentheses of the original expression. Applying the product rule is the easy part. So, I think what he is saying is let u = everything which is inside the parentheses, then we have y = u^3, dy/du = 3 u^2 = 3 times (everything inside the parentheses) squared. ![]() Where Sal draws a parenthesis and says something goes in there seems to gloss over a little bit how the chain rule entity is identified and then goes in there, and next he says to apply the product rule. The video is about applying the chain rule twice, there may be other ways to get the answer, but first I want to understand how to apply the chain rule twice, which can be confusing.
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