Final Answer:
f'(0) = e⁰ . g'(0) + e⁰ . g(0) = 7
Step-by-step explanation:
To find the derivative f'(x), which represents the rate of change of the function f(x) = eˣ . g(x), at the point x = 0, we use the product rule of derivatives. This rule states that the derivative of a product of two functions u and v is given by u'v + uv'.
Given that u = eˣ and v = g(x), we differentiate u to get u' = eˣ and substitute the given values for g(0) = 3 and g'(0) = 4. Plugging these values into the derivative expression at x = 0 yields:
f'(0) = e⁰ . 4 + e⁰ . 3 = 4 + 3 = 7.
Therefore, the derivative of f(x) at x = 0 (i.e., f'(0) is 7.
Understanding derivatives and their applications in calculus is crucial for analyzing how functions change over their domains. One fundamental rule in calculus, the product rule, is used to differentiate the product of two functions.