High School

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Given the function [tex]y = (-10x^3 + 6x^{-2} + 8)(7x^3 + 3)[/tex], find [tex]\frac{dy}{dx}[/tex].

A) 0
B) 1
C) -10x^3 + 6x^{-2} + 8
D) 70x^4 - 18x

Answer :

Final answer:

To find the derivative of the function y=(-10x^3+6x^{-2}+8)(7x^3+3), the product rule of differentiation must be applied, resulting in dy/dx = 70x^5 - 20x^2 - 18x^{-3}.

Explanation:

The student is asking for the derivative of the function y=(-10x^3+6x^{-2}+8)(7x^3+3). To find dy/dx, we need to apply the product rule of differentiation, which states that the derivative of a 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.

First, we differentiate each part:

  • d/dx(-10x^3) = -30x^2
  • d/dx(6x^{-2}) = -12x^{-3}
  • d/dx(8) = 0
  • d/dx(7x^3) = 21x^2
  • d/dx(3) = 0

Now, let's apply the product rule:

d/dx(y) = (-10x^3+6x^{-2}+8)(21x^2) + (-30x^2+(-12x^{-3}))(7x^3+3)

After simplifying, we will get the answer dy/dx = 70x^5 - 20x^2 - 18x^{-3}.

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