Final answer:
To find f'(x) of f(x) = x^3 · e^(8x), we apply the product rule: differentiate each function separately and then use the formula u'(x)v(x) + u(x)v'(x) to obtain 3x^2e^(8x) + 8x^3e^(8x).
Step-by-step explanation:
To compute the derivative f′(x) of the function f(x)=x³·e⁸⁰x, we need to apply the product rule of differentiation. This rule states that the derivative of a product of two functions is given by the derivative of the first function times the second function, plus the first function times the derivative of the second function.
So, let's let u(x) = x³ and v(x) = e⁸⁰x. The derivative of u with respect to x is u′(x) = 3x², and the derivative of v with respect to x is v′(x) = e⁸⁰x·8, since the derivative of e⁰x with respect to x is e⁰x, and then we multiply by the constant 8 due to the chain rule.
Applying the product rule:
f′(x) = u′(x)·v(x) + u(x)·v′(x)
f′(x) = (3x²)·(e⁸⁰x) + (x³)·(e⁸⁰x·8)
f′(x) = 3x²e⁸⁰x + 8x³e⁸⁰x
This result is the derivative of the given function.