Final answer:
To find the derivative of f(x) = x^{3}(2x+1)(x^{2}-3), use the product rule, which involves finding the derivatives of x^{3}(2x+1) and (x^{2}-3) and then applying the rule to get f'(x).
Step-by-step explanation:
The student has asked to find the derivative of the function f(x) = x^{3}(2x+1)(x^{2}-3). To compute the derivative, we use the product rule of differentiation, which states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. The function f(x) can be viewed as a product of two functions, u(x) = x^{3}(2x+1) and v(x) = (x^{2}-3). Applying the product rule, we have:
Derivative of f(x)
f'(x) = u'(x)v(x) + u(x)v'(x)
First, find the derivatives:
- u'(x) = 3x^{2}(2x+1) + x^{3}(2)
- v'(x) = 2x
Now multiply and add according to the product rule:
f'(x) = (3x^{2}(2x+1) + 2x^{4})(x^{2}-3) + (x^{3}(2x+1))(2x)
Then, expand and simplify to obtain the final derivative.