If f(x)=6x+9/3x+1
then f'(x)_____?
and f'(3)=_____?
We will be happy to critique your thinking.
f'(x) = -21/(3x + 1)^2
To find the derivative of the function f(x), denoted as f'(x), we can use the power rule and the quotient rule.
First, let's rewrite the function f(x):
f(x) = (6x + 9)/(3x + 1)
To differentiate this function, we will first apply the quotient rule, which can be stated as follows:
If f(x) = g(x)/h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2
Let's apply the quotient rule to f(x):
g(x) = 6x + 9
g'(x) = 6 (derivative of 6x + 9 with respect to x)
h(x) = 3x + 1
h'(x) = 3 (derivative of 3x + 1 with respect to x)
Now, let's substitute these values into the quotient rule:
f'(x) = [(6)(3x + 1) - (6x + 9)(3)] / (3x + 1)^2
Simplifying this expression:
f'(x) = (18x + 6 - 18x - 27) / (3x + 1)^2
f'(x) = (-21) / (3x + 1)^2
f'(x) = -21 / (3x + 1)^2
So, the derivative of f(x), or f'(x), is -21 / (3x + 1)^2.
Now, to find f'(3), we can substitute x = 3 into the expression we found for f'(x):
f'(3) = -21 / (3(3) + 1)^2
f'(3) = -21 / (9 + 1)^2
f'(3) = -21 / 10^2
f'(3) = -21 / 100
f'(3) = -0.21
Therefore, f'(3) = -0.21.