How do you use the chain rule to differentiate #y=(x^2+5x-2)^2#?
1 Answer
Jul 5, 2017
We could use the chain rule via an explicit substitution:
# u= x^2+5x-2 => (du)/dx = 2x + 5#
# y=u^2 => dy/(du) = 2u #
# dy/dx = dy/(du) * (du)/dx = 2(x^2+5x-2)(2x+5) #
In practice the substitution can be done implicit so we can just write the answer down using
# d/dx G(x)^n = nG(x)^(n-1) * (dG)/dx #
which comes directly from the chain rule, so with practice we can just write down the derivative as
# dy/dx = 2(x^2+5x-2)^1 d/dx (x^2+5x-2) #
# \ \ \ \ \ = 2(x^2+5x-2)(2x+5) # , as above
