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Math 1210: Homework Week 5 Name: Theorem 1 (Chain Rule). Suppose g is differentiable at x and f is differentiable at g(x). Then fog is differentiable at x and (fog)'(x) = f'(g(x))g'(x), or in operator notation, Da(f(g(x)) = f'(g(x))g'(x). Remark. The chain rule takes an interesting and suggestive form in Liebniz notation. Specifically, we set u = g(x) and y = f(u) = f(g(x)). Now the chain rule looks like dy dy du dx du da So we see that the chain rule is suggested by imagining that we can cancel the du terms above. Exercise 1. Suppose f(x) = x2 and g(x) = x3 - x. Use the Chain Rule to find Dx(f(g(x)). Compare this to what you get when you first find f(g(x)) by substitution then differentiate that