Which of the following corresponds to the derivative of f(x)=xe^x, using the primary definition of a derivative, reduced to its simplest form before taking the limit?
wrong. We all know that
f'(x) = e^x + xe^x
Apologies for the mistake. You are correct.
The derivative of f(x) = xe^x can be calculated using the product rule and the chain rule:
f'(x) = (1)(e^x) + (x)(e^x) = e^x + xe^x.
Thus, the correct derivative of f(x) = xe^x is:
f'(x) = e^x + xe^x.
To find the derivative of f(x) = xe^x using the definition of the derivative, we can use the formula:
f'(x) = lim(h→0) [f(x + h) - f(x)] / h
Let's apply this formula to f(x) = xe^x:
f'(x) = lim(h→0) [(x + h)e^(x + h) - xe^x] / h
Expanding the terms and simplifying:
f'(x) = lim(h→0) [x*e^x*e^h + h*e^x*e^h - xe^x] / h
f'(x) = lim(h→0) [(x*e^x + h*e^x + h*e^x*e^h - xe^x)] / h
Now, let's reduce this expression to its simplest form before taking the limit:
f'(x) = lim(h→0) [x*e^x - xe^x + h*e^x + h*e^x*e^h] / h
f'(x) = lim(h→0) [(e^x - e^x)*x + h*e^x + h*e^x*e^h] / h
f'(x) = lim(h→0) [h*e^x + h*e^x*e^h] / h
Reducing the common factor of h:
f'(x) = lim(h→0) [h*(e^x + e^x*e^h)] / h
f'(x) = lim(h→0) (e^x + e^x*e^h)
Taking the limit as h approaches 0:
f'(x) = e^x + e^x * e^0
Since e^0 = 1, the derivative of f(x) = xe^x reduced to its simplest form before taking the limit is:
f'(x) = e^x + e^x * 1
f'(x) = e^x + e^x
Therefore, the derivative of f(x) = xe^x using the primary definition of the derivative, reduced to its simplest form before taking the limit, is:
f'(x) = 2e^x