How do you use the chain rule to differentiate #y=(x^2+5x-2)^2#?

Question

Christopher Agresta · Accepted Answer

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

Luke Alvarez · Answer

undefined To differentiate y=(x^2+5x-2)^2 using the chain rule, first identify the outer function, which is the square function, and the inner function, which is the expression (x^2+5x-2). Then, apply the chain rule, which states that the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to x, gives the derivative of the composite function.

1. Derivative of the outer function: (2u), where u is the inner function.
2. Derivative of the inner function: (2x+5).

Therefore, the derivative of y=(x^2+5x-2)^2 with respect to x is:

dy/dx = 2(x^2+5x-2)(2x+5)