High School

Use the Product Rule to find the derivative of the given function.

Function: [tex]F(x) = 9x^4 (x^2 - 9x)[/tex]

a) Use the Product Rule to find the derivative of the function.

b) Find the derivative by multiplying the expressions first.

Select the correct answer below:

A. The derivative is [tex]9x^4 (x^2 - 9x)[/tex].

B. The derivative is [tex]9x^4 + (x^2 - 9x)[/tex].

C. The derivative is [tex]9x^4 + 108x^2[/tex].

D. The derivative is [tex](x^2 - 9x)()[/tex].

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.

To know more about derivative, visit:

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