find the derivative of the function:
f(x)=6ln(x)/x^9
f = 6ln(x) * x^-9
f' = 6/x - 9x^-10
or, 3/x (2-3/x^9)
Steve, we are looking at a product rule.
f'(x) = 6ln x (-9x^-10) + x^-9 (6/x)
= -54lnx/x^10 + 6/x^10
or
= 6/x^10 (-9lnx + 1)
Good catch, Reiny.
What was I thinking?!?
Sorry, Tara. You must have been scratching your head on that one.
As Bugs Bunny was wont to say:
That's pretty good, Doc, but it ain't the way I heered it!
Thank you, and I was a bit confused. Thanks though!
To find the derivative of the function f(x) = 6ln(x)/x^9, we can use the quotient rule of differentiation. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), where both g(x) and h(x) are differentiable functions, then the derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's apply the quotient rule to find the derivative of f(x) = 6ln(x)/x^9:
1. First, let's find g'(x) and h'(x):
g(x) = 6ln(x)
g'(x) = derivative of 6ln(x)
We can use the chain rule to find the derivative of ln(x). The derivative of ln(x) is 1/x. Therefore:
g'(x) = 6 * 1/x = 6/x
h(x) = x^9
h'(x) = derivative of x^9
The power rule tells us that the derivative of x^n is n*x^(n-1). Therefore:
h'(x) = 9 * x^(9-1) = 9x^8
2. Now we can substitute g(x), g'(x), h(x), and h'(x) into the quotient rule formula to find f'(x):
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
f'(x) = (6/x * x^9) - (6ln(x) * 9x^8) / (x^9)^2
Simplifying further:
f'(x) = (6x^8) - (54x^8ln(x)) / x^18
Finally, we can simplify the expression:
f'(x) = (6x^8(1 - 9ln(x))) / x^18
Simplifying the expression further:
f'(x) = (6(1 - 9ln(x))) / x^10
Therefore, the derivative of the function f(x) = 6ln(x)/x^9 is f'(x) = (6(1 - 9ln(x))) / x^10.