Question

The derivative of a function f is given by f'(x) = 0.1xe⁰.²⁵ˣ. What is the function f(x)?

a) 0.04e⁰.²⁵ˣ + C
b) 0.1e⁰.²⁵ˣ + C
c) 0.02e⁰.²⁵ˣ + C
d) 0.5e⁰.²⁵ˣ + C

Answer 1

To find the original function from its derivative f'(x) = 0.1xe⁰.²⁵ˣ, we integrate to get the antiderivative. The correct original function is 0.4e⁰.⁴ˣ⁵ + C, which corresponds to option (a). The solution process involves the reverse of the power rule for exponential functions.

Step-by-step explanation:

The equation given for the derivative of a function, f'(x) = 0.1xe⁰.²⁵ˣ, indicates that we need to find the antiderivative to determine the original function f(x). Upon integrating this derivative, the integral procedure for such a function requires recognizing that the variable x is a factor of the exponential function.

Using the reverse of the power rule for exponential functions, we know that the integral of e⁰.⁴ˣ⁵ with respect to x is e⁰.⁴ˣ⁵ divided by the coefficient of x in the exponent when the base is e. This yields the integral 0.4xe⁰.⁴ˣ⁵. However, we also need to account for the integration constant, commonly designated as C. Therefore, the function f(x) that corresponds to the given derivative is 0.4e⁰.⁴ˣ⁵ + C, which corresponds to option (a).