[MUSIC] The chain rule is based on the fact that y is a function of u and that u is a function of x. This makes y be what we call a composite function of x. Suppose that x changes, this leads to a two stage chain reaction. Firstly, u reacts directly to the change in x and second, it will be y, which reacts in turn to this change in u. Now, if we know the rate of change of du/dx, and dy/du. What is the rate of change, dy/dx? The relationship within this rate of change is quite simple. Dy/dx = dy/du x du/dx. More formally, we say that if y is a differentiable function of u, and u is a differentiable function of x, then y is a differentiable function of x, and the above formula holds. A special k is when y = u to the power of a. Then dy/du = au to the power of a- 1. Then the chain rule gets a generalized power rule. Y = u to the power of a. y prime = au to the power of a- 1 u prime. Where (u = g(x)) The chain rule is very powerful and useful in practical contests. Using the Leibniz's notation makes easy to remember the chain rule. On the left hand side of our formula, we have the results of canceling the derivatives of u on the right hand side. It is important to keep in mind that dy/du and the du/dx are not fractions. They are symbols for derivatives. Also, the derivative of u is not a number, therefore, canceling is not defined. For example, if we need to find dy/dx when y = u to the power of 4, and u = 1- x to the power of 2. Here, we can use directly our formula. Since the dy/du = 4u to the power of 3. And du/dx = -2x. We have dy/dx = dy/du x du/dx = 4u to the power of 3 (-2x) = 8xu to the power of 3 = 8x (1- x to the power of 2), everything to the power of 3. If we need to find dy/dx when y = 3u to the power of 2- 2u- 2 and u = x to the power of 3 + 5. The solution is dy/dx = dy/du x du/dx = d/du (y = 3u to the power of 2- 2u- 2) x d/dx(x to the power of 3 + 5). Firstly, we take the first derivative, and we get the following result which is (6u- 2) x (3x). Now we substitute u. And therefore we have dy/dx = [6(x to the power of 3 + 5)- 2] x (3x) = [6x to the power of 3 + 28] x (3x) = 18x to the power of 4 + 84x. [MUSIC]