derivative questions! HELP ME?

Question

if Iam given a function g(x) = absolute value of(x *e^x)

what is the derivative of that function??

Answer 1

First, we must determine where it's negative or nonnegative, since |x| = x if x >= 0, and -x otherwise.

e^x > 0 for all x, so g(x) >= 0 if and only if x >= 0.

So, we write the function as

g(x) = xe^x if x >= 0, and -xe^x if x < 0.

For x > 0, we have g'(x) = e^x + xe^x, by the product rule.

Similarly, for x < 0, we have g'(x) = -e^x - xe^x.

At x=0, we have a problem: the derivative may not exist. To determine this, we compute the left-hand derivative at x=0, by evaluating -e^x - xe^x at x=0. This is the limit of g'(x) as x -> 0 from the left. We get -1.

For the right-hand derivative, we evaluate e^x + xe^x at x=0, and obtain the limit of g'(x) as x->0 from the right. We get 1.

Since these don't agree, g(x) is not differentiable at x=0.

Answer 2

y = xe^x
using a substitution, y = uv. this is the product rule, to find y' (the 1st derivative of y) you do y' = v*u' + u*v'
u = x u' = 1
v = e^x v' = e^x
so y' = 1*e^x + x*e^x
y' = e^x(1+x)

Answer 3

g'(x) = x e^x + e^x

g'(x) = e^x (x + 1)