# How do you differentiate #g(u) =(u^-2 + u^3)(u^5 - u^-2) # using the product rule?

- Product rule:
#(ab)'=a'b+ab'#

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To differentiate the function g(u) = (u^-2 + u^3)(u^5 - u^-2) using the product rule, follow these steps:

- Identify the two functions being multiplied: f(u) = u^-2 + u^3 and g(u) = u^5 - u^-2.
- Apply the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
- Find the derivatives of the individual functions: f'(u) = (-2u^-3 + 3u^2) and g'(u) = (5u^4 + 2u^-3).
- Apply the product rule formula: g'(u) = f(u)g'(u) + g(u)f'(u).
- Substitute the derivatives and the original functions into the formula: g'(u) = (u^-2 + u^3)(5u^4 + 2u^-3) + (u^5 - u^-2)(-2u^-3 + 3u^2).
- Simplify the expression to get the final result: g'(u) = 5u^2 - 2u^-5 + 2u^8 - 3u^2 - 10u^3 + 3u^-1.

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