would the derivative of this be 0/1-1/0?
problem: 5/x-x/6
To find the derivative of the function 5/x - x/6, we will need to use the quotient rule.
The quotient rule states that if we have a function that is the quotient of two other functions, say f(x) = g(x) / h(x), then the derivative of f(x) can be found as follows:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In our case, g(x) = 5 and h(x) = x - x/6 = 6x/6 - x/6 = (6x - x)/6 = 5x/6. Let's differentiate g(x) and h(x) to find g'(x) and h'(x).
g'(x) is the derivative of 5, which is 0.
h'(x) is the derivative of 5x/6. To find this, we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = cx^n, then its derivative is given by f'(x) = c * n * x^(n-1).
In our case, c = 5/6 and n = 1. Applying the power rule, we get:
h'(x) = (5/6) * 1 * x^(1-1) = (5/6) * x^0 = 5/6
Now, let's substitute the values we found into the quotient rule equation:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= (0 * (5x/6) - 5 * (5/6)) / ((5x/6))^2
= (-25/6) / (25x^2/36)
= -25/6 * 36/(25x^2)
= -6/6 * 36/(5x^2)
= -36/(5x^2)
So, the derivative of 5/x - x/6 is -36/(5x^2).