What is the chain rule?

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Answer 1 · 2015-11-03T22:23:38

The chain rule for differentiation is essentially:

#(dy)/(dx) = (dy)/(du) * (du)/(dx)#

So if #y = f(g(x))# then #d/(dx) y = f'(g(x)) * g'(x)#

For example, suppose #y = (x^2+x-1)^10#

Let #u = x^2+x-1#

Then #y = u^10# and using the power rule:

#(dy)/(du) = 10 u^9#

#(du)/(dx) = 2x+1#

Hence:

#(dy)/(dx) = (dy)/(du) * (du)/(dx) = 10u^9 * (2x+1) = 10(x^2+x-1)^9(2x+1)#

Or we could formulate this as follows:

#g(x) = x^2+x-1#

#f(u) = u^10#

#y = f(g(x)) = (x^2+x-1)^10#

#d/(dx) y = f'(g(x)) * g'(x) = 10(g(x))^9 * (2x+1)#

#=10(x^2+x+1)^9(2x+1)#