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Answer :
The derivative is 9x4+108x2 so option (C) is the correct answer.
The given function is F(x)=9x⁴(x²−9x). The product rule is a formula to differentiate the product of two functions.
If a and b are two differentiable functions, then the product rule states that d/dx(a*b) = a*d/dx(b) + b*d/dx
(a) Here, we have a product of two functions, and we will use the product rule to find the derivative of the given function. Therefore, we have f(x) = 9x⁴(x²−9x) = u*v
where u = 9x⁴ and
v = (x²−9x)
Now, we can apply the product rule to find the derivative of f(x) as follows: f'(x) = u'v + uv'
where u' is the derivative of u with respect to x, and v' is the derivative of v with respect to x.
Using the product rule, we have f'(x) = (36x³)(x²−9x) + 9x⁴(2x−9)
f'(x) = 36x³(x²−9x) + 18x⁵−81x⁴
Thus, the derivative of the given function is: f'(x) = 36x³(x²−9x) + 18x⁵−81x⁴
Therefore, the derivative is 9x4+108x2 is the correct answer.
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