Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. The chain rule. If F of U is differentiable at the point U equal G of X&G of X is differentiable at X, then the composite function F of G of X = F of G of X is differentiable at X&F of G of prime of X = F prime G of X * g prime of X. This is sometimes referred to the inside outside rule. So we take the derivative of the outside portion, leaving the inside alone times the derivative of the inside portion. In Lebanese notation, if Y equal F of U&U equal G of X, then the derivative of Y in terms of X is just the derivative of Y in terms of U times the derivative of U in terms of X. So it's really easier to see the chain rule if we do some examples. If we start with Y equal 4 -, 3 X to the 9th, we're going to do the derivative of the outside portion first. So we're going to bring down the 9, take it to the one less power, and leave the inside alone. Then we're going to take the derivative of the inside. Well, the derivative of that 4 -, 3 X is just -3. So the derivative of Y equal the quantity 4 - 3 X to the 9th is going to be Y prime equaling -27 the quantity 4 - 3 X to the 8th. So we did the derivative of the outside portion, leaving the inside alone times the derivative of the inside. Another example, Y equal cotangent of π -, 1 / X. So we're going to do the derivative of the outside. The derivative of the cotangent is negative cosecant squared. We leave that π - 1 / X alone. But now we need to take the derivative of that π - 1 / X while the derivative of Pi π is a constant. So the derivative of constant 0, the derivative of -1 / X. We could think of it as negative X to the -1 and use the power rule so bring down that negative, making it a + X to the -2 or 1 / X ^2. So Y prime is just going to equal -1 / X ^2 cosecant squared π -, 1 / X. When we have trig functions in a monomial or in a term, we usually write the trig at the very end. Another example, R = Theta squared times cosine 2 Theta. So now we're going to use the product rule and the chain rule. So we need to figure out the derivative of sine of Theta squared times cosine 2 Theta plus the derivative of the cosine 2 Theta times sine of Theta squared. So the derivative of sine of Theta squared, we've got to do the chain rule. We're going to do the outside. The derivative of sine was cosine, so it's cosine Theta squared. Now we need to do the derivative of the inside. The derivative of Theta squared is 2 Theta and that's still going to get multiplied by the cosine 2 Theta plus going to do the chain rule. Again, the derivative of the cosine is negative sine, leaving the inside alone. So negative sine 2 Theta then times the derivative of the inside. The derivative of two Theta is 2, and that's all multiplied by the first term, the sine Theta squared. So when we simplify this up, we get 2 Theta cosine Theta squared, cosine 2 Theta -2 sine 2 Theta sine Theta squared. Thank you and have a wonderful day.