Question

If g(x) = ((x+1)eˣ)/(x-2), find g'(0).
a. 0
b. 1
c. -1
d. e

Answer 1

To find out the derivative of the function g(x) at x=0, apply the quotient rule for derivatives. After finding the derivatives for the numerator and denominator, evaluate them at x=0. The derivative g'(0) turns out to be the constant e.

Step-by-step explanation:

The given function is g(x) = \frac{(x+1)e^x}{x-2} and we are asked to find its derivative at x=0, denoted as g'(0).

To find the derivative of g(x), we need to use the quotient rule which is given by:

\((f/g)' = \frac{f'g - fg'}{g^2}\), where f(x) = (x+1)e^x and g(x) = x-2.

1. First, find the derivative f'(x) = e^x + (x+1)e^x using the product rule.
2. Then, calculate the derivative g'(x) = 1.
3. Next, substitute these derivatives and the original functions into the quotient rule formula.
4. Finally, evaluate the derivative at x=0 to find g'(0).

The derivative of g(x) at x=0 is g'(0) = e. Therefore, the correct answer is (d) e.