Answer :
Final answer:
The derivative of the function y = (x^2 - 3)(x^3 - 6) is calculated using the product rule to be y' = 5x^4 - 9x^2 - 12x, which does not match any of the provided options in the question. None of the option is correct.
Explanation:
To find the derivative of the function y = (x2 - 3)(x3 - 6) using the product rule, we first identify the two functions that are being multiplied together. Let u(x) = x2 - 3 and v(x) = x3 - 6. The product rule states that the derivative of a product of two functions is given by u'v + uv', where u' and v' are the derivatives of u and v, respectively.
Step 1: Differentiate u(x) and v(x).
u'(x) = 2x
v'(x) = 3x2
Step 2: Apply the product rule.
y' = u'v + uv'
y' = (2x)(x3 - 6) + (x2 - 3)(3x2)
Step 3: Simplify the expression.
y' = 2x4 - 12x + 3x4 - 9x2
y' = 5x4 - 9x2 - 12x
Therefore, the derivative of the function is 5x4 - 9x2 - 12x, which is not one of the options provided in the question. It seems there might have been a mistake in the options or a typo in the question.
