# Question #96976

##### 1 Answer

#### Explanation:

For differentiable functions

# = lim_(hrarr0) ([f(x+h)g(x+h)]- [f(x)g(x+h)] + [f(x)g(x+h)]- [f(x)g(x)])/h#

# = lim_(hrarr0) ([f(x+h)g(x+h)]- [f(x)g(x+h)])/h + ([f(x)g(x+h)]- [f(x)g(x)])/h#

# = lim_(hrarr0) (f(x+h)- f(x)]/h g(x+h) + f(x)(g(x+h) - g(x))/h#

# = lim_(hrarr0) (f(x+h)- f(x)]/h lim_(hrarr0)g(x+h) + lim_(hrarr0)f(x) lim_(hrarr0)(g(x+h) - g(x))/h#

# = d/dx[f(x)] g(x)+f(x)d/dx[g(x)]#

# = f'(x)g(x)+f(x)g'(x)#

**Applied to #xe^x#**

# = 1*e^x + x e^x#

**In the function for this question,** we have a quotient

We'll be using the quotient rule, so we need

# = (-[(e^x+xe^x)(x+e^x) + (1-xe^x)(1+e^x)])/(x+e^x)^2#

Expand and simplify to get: