Partial derivatives answer?

Question

z = (x^2 + 5x - 2y)^9

∂z / ∂x = ???

Answer 1

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z = (x² + 5x - 2y)⁹

To get dz/dx, with other variables in place (y) we treat that as a constant.

We will need a chain rule so we can set that up below:

z = u⁹ and u = x² + 5x - 2y

Getting the derivative of each:

dz/du = 9u⁸ and du/dx = 2x + 5 (the constant term drops off)

so:

dz/dx = dz/du * du/dx

dz/dx = 9u⁸ (2x + 5)

Substituting the expression in terms of x and y instead of u:

dz/dx = 9(x² + 5x - 2y)⁸ (2x + 5)