In other words, it helps us differentiate composite functions. Its like looking inside a clock and saying. The chain rule states that the derivative of f(g(x)) is f(g(x))g(x). The next two examples illustrate 'functional' and 'Leibniz' methods of attacking the same problem using the chain rule. The chain rule is about going deeper into a single part (like f) and seeing if its controlled by another variable. However, a fully rigorous proof is beyond the secondary school level. The proof above is not entirely rigorous: for instance, if there are values of \(\Delta x\) close to zero such that \(g(x+\Delta x) - g(x) = 0\), then we have division by zero in the first limit. the derivative of a quotient \(\dfrac \bigl = f'(g(x))\,g'(x),.the derivative of a product \(f(x)\,g(x)\) is not the product of the derivatives The chain rule allows us to differentiate a function that contains another function.the derivative of a difference is the difference of the derivatives.the derivative of a sum is the sum of the derivatives.If we dont take derivatives to find the minimum-maybe because f is a function of other functions. The first is 'application,' the second looks like 'theory.' If we minimize f to save time or money or energy, that is an application. the derivative of a constant multiple is the constant multiple of the derivative 13.5 The Chain Rule Calculus goes back and forth between solving problems and getting ready for harder problems.
We now move to some more involved properties of differentiation. The chain rule is a method to compute the derivative of the functional composition of two or more functions.
Content The product, quotient and chain rules
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