Find f'(a) for
f(x)=13x/3x+1
f'= 13/(3x+1) - 13x*3/(3x+1)^2
check that carefully.
If f=u*v^-1
then f'=u' * v^-1 -u*v^-1 * v'
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To find f'(a), we need to find the derivative of the function f(x) with respect to x and then evaluate it at x = a.
Given that f(x) = (13x) / (3x + 1), we'll start by using the quotient rule to find the derivative.
The quotient rule states that the derivative of f(x) = u(x) / v(x) is given by:
f'(x) = [u'(x) * v(x) - u(x) * v'(x)] / [v(x)]^2
Applying the quotient rule to f(x) = (13x) / (3x + 1), we get:
f'(x) = [(13 * (3x + 1)) - (13x * 3)] / [(3x + 1)^2]
Now, simplifying the expression, we get:
f'(x) = (39x + 13 - 39x) / (9x^2 + 6x + 1)
f'(x) = 13 / (9x^2 + 6x + 1)
To find f'(a), we substitute x = a back into the derivative expression:
f'(a) = 13 / (9a^2 + 6a + 1)
Therefore, f'(a) = 13 / (9a^2 + 6a + 1).