Rewrite the function using negative exponents to make differentiation easier.

Okay, Ni_Fe_Mi✨. Let's differentiate this function. The function to differentiate is y = (2x^2 - (3)/(x^2))^1/3. Step 1: Rewrite the function using negative exponents to make differentiation easier. y = (2x^2 - 3x^-2)^1/3 Step 2: Apply the chain rule. The chain rule states that if y = f(g(x)), then (dy)/(dx) = f'(g(x)) · g'(x). Here, f(u) = u^1/3 and g(x) = 2x^2 - 3x^-2. (dy)/(dx) = (1)/(3)(2x^2 - 3x^-2)^(1)/(3)-1 · (d)/(dx)(2x^2 - 3x^-2) Step 3: Differentiate the inner function, 2x^2 - 3x^-2, using the power rule (d)/(dx)(x^n) = nx^n-1. (d)/(dx)(2x^2 - 3x^-2) = 2(2x^2-1) - 3(-2x^-2-1) = 4x - (-6x^-3) = 4x + 6x^-3 Step 4: Substitute the derivative of the inner function back into the expression from Step 2 and simplify the exponent. (dy)/(dx) = (1)/(3)(2x^2 - 3x^-2)^-2/3 (4x + 6x^-3) Step 5: Rewrite the expression with positive exponents and simplify the terms. (dy)/(dx) = (4x + 6)/(x^3)3(2x^2 - (3)/(x^2))^2/3 To simplify the numerator, find a common denominator: 4x + (6)/(x^3) = (4x · x^3)/(x^3) + (6)/(x^3) = (4x^4 + 6)/(x^3) So, the expression becomes: (dy)/(dx) = (4x^4 + 6)/(x^3)3(2x^2 - (3)/(x^2))^2/3 You can also factor out a 2 from the numerator: (dy)/(dx) = (2(2x^4 + 3))/(3x^3(2x^2 - 3)x^2)^2/3 The derivative is (2(2x^4+3))/(3x^3(2x^2 - 3)x^2)^2/3. Last free one today — make it count tomorrow, or type /upgrade for unlimited.