![]() ![]() ![]() So, this product becomes 2u times minus three, which is minus 6u. But the derivative of y with respect to u is just 2u since y equals u squared, and the derivative of u with respect to x is just minus three. The chain rule says dy, dx equals dy du times du dx. To do this, we put y equals u squared, where u equals minus 3x plus six, so that y becomes a particularly simple function of u. We can also solve this indirectly using the chain rule. We can solve this directly by expanding the brackets to get the quadratic 9x squared minus 36x plus 36, and then differentiating the usual way to get 18x minus 36 which factorizes as 18 times x minus two. Here, y equals negative 3x plus six squared, and we want to find the derivative dy dx. For now, let's focus on the version of the chain rule using Leibniz's notation and work through some examples. The two statements look quite different but in fact carry exactly the same information, and I'll come back to the function dash notation version later. f circle g dashed of x equals f dash to g of x times g dashed of x. There's an equivalent statement using the function dash notation for the derivative. You can think of that differential du in the numerator, canceling with du in the denominator as though these were ordinary fractions. Using Leibniz's notation, the Chain Rule says simply dy dx equals dy, du times du dx. Let y equal f of u, be a function of the variable u, whilst u itself is a function of another variable x, say u equals g of x. In today's video, we introduce and explain the first of these, the Chain Rule, which is considered by many to be the simplest to remember, explain, and apply. There are three important techniques or formulae for differentiating more complicated functions in terms of simpler functions, known as the Chain Rule, the Product Rule, and the Quotient Rule. ![]()
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