Final answer:
The function f(x) = x^3(x + 8) is differentiated by applying the product rule, yielding f'(x) = 4x^3 + 24x^2. This process involves finding derivatives of individual terms and applying the rule that states the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function.
Step-by-step explanation:
How to Differentiate the Function f(x) = x^3(x + 8)
Differentiating the function f(x) = x^3(x + 8) involves applying the product rule because the function is the product of two functions of x, which are x^3 and (x + 8). According to the product rule, if we have two functions u(x) and v(x), their derivative u'(x)v(x) + u(x)v'(x). Here, we let u(x) = x^3 and v(x) = x + 8, then:
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- The derivative of u(x) with respect to x is u'(x) = 3x^2.
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- The derivative of v(x) with respect to x is v'(x) = 1.
By applying the product rule, we get:
f'(x) = u'(x)v(x) + u(x)v'(x) = 3x^2(x + 8) + x^3(1) = 3x^2x + 24x^2 + x^3 = 4x^3 + 24x^2
The differentiated function is f'(x) = 4x^3 + 24x^2. This process shows the application of the product rule and the basic principles of differentiation.