Let f(x)=3x^2+x+3/2x+1. What is the derivative of f with respect to x
To find the derivative of the function f(x) = 3x^2 + x + 3/(2x + 1) with respect to x, we can apply the quotient rule and power rule.
The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then its derivative can be found using the following formula:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's apply this to our function f(x) = 3x^2 + x + 3/(2x + 1):
g(x) = 3x^2 + x
h(x) = 2x + 1
g'(x) = 6x + 1 (using the power rule to differentiate g(x) = 3x^2 + x)
h'(x) = 2 (using the power rule to differentiate h(x) = 2x + 1)
Now, substitute these values into the quotient rule:
f'(x) = ((6x + 1) * (2x + 1) - (3x^2 + x) * 2) / (2x + 1)^2
Simplifying the equation further:
f'(x) = (12x^2 + 8x + 1 - 6x^2 - 2x) / (2x + 1)^2
= (6x^2 + 6x + 1) / (2x + 1)^2
Therefore, the derivative of f(x) with respect to x is given by f'(x) = (6x^2 + 6x + 1) / (2x + 1)^2.