How is the chain rule different from the product rule?

1 Answer
Jim H
Mar 14, 2016

The chain rule is used to differentiate compositions of functions. The product rule is used to differentiate products of function.

Explanation:

Composition and product are different operations.

For #f(x) = 2x+3# and #g(x) = 5x+7#, the composition

#(f@g)(x) = f(g(x)) = f(5x+7) = 2(5x+7)+3 = 10x + 17#

is not at all the same as the product

#(fg)(x) = f(x)g(x) = (2x+3)(5x+7) = 10x^2 +29x+21 #.

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