Find the derivative of the function. g(x) = (e^x)/(3 + 3x)
e^x /(3+3x)
g'(x) = xe^x/(3(x+1)^2
To find the derivative of the function g(x) = (e^x)/(3 + 3x), we can use the quotient rule.
The quotient rule states that if we have a function in the form f(x) = u(x)/v(x), where u(x) and v(x) are differentiable functions, then the derivative of f(x) is given by the formula:
f'(x) = (v(x)*u'(x) - u(x)*v'(x))/(v(x))^2
Let's apply this rule to our function.
First, we need to find u(x) and v(x). In this case, u(x) = e^x and v(x) = 3 + 3x.
Now, we need to find the derivatives of u(x) and v(x):
u'(x) = d/dx(e^x) = e^x
v'(x) = d/dx(3 + 3x) = 3.
Now, we can substitute these values into the quotient rule formula:
g'(x) = [(3 + 3x)(e^x) - (e^x)(3)] / (3 + 3x)^2
Simplifying this expression:
g'(x) = [3e^x + 3xe^x - 3e^x] / (3 + 3x)^2
g'(x) = [3xe^x] / (3 + 3x)^2
Therefore, the derivative of g(x) is g'(x) = [3xe^x] / (3 + 3x)^2.