Answer :
Final answer:
The correct differentiation of the function [tex]y=2x(8x^3-3x)[/tex] is [tex]64x^3 - 12x[/tex], using the product rule for differentiation. The answer is option D.
Explanation:
The question is asking to differentiate the function [tex]y=2x(8x^3-3x)[/tex]. To solve this, we apply the product rule of differentiation, which states that if you have a function u(x)v(x), then its derivative, [tex]d/dx[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)[/tex]. Here, our u(x) = 2x and v(x) = [tex]8x^3 - 3x[/tex].
First, we find the derivatives of u(x) and v(x):
- u'(x) = 2
- v'(x) = [tex]24x^2 - 3[/tex]
Applying the product rule:
[tex]d/dx[u(x)v(x)] = 2(8x^3 - 3x) + 2x(24x^2 - 3)[/tex]
Simplifying, we get:
[tex]d/dx = 16x^3 - 6x + 48x^3 - 6x[/tex]
[tex]d/dx = 64x^3 - 12x[/tex], which matches option D.
