find the derivative of ln(sqrt((1-ln(x))/(1+ln(x)))
It looks messy, but you have
y = ln(sqrt((1-ln(x)) - ln(1+ln(x)))
= 1/2[ ln(1-lnx) - ln(1+lnx) ]
Now, using the chain rule, that is
y' = 1/2 [ 1/(1-lnx)*(-1/x) - 1/(1+lnx)*(1/x) ]
= 1/2 * 1/(x(lnx-1)(lnx+1))
In google paste:
math10 derivative calculator
Whe you see list of results click on:
Free Step-by-Step First Derivative Calculator(Solver) - Math10
When page be open in rectangle Function paste:
ln(sqrt((1-ln(x))/(1+ln(x))))
and click option:
Find derivative
You will see solution step-by-step.
To find the derivative of ln(sqrt((1-ln(x))/(1+ln(x))), we can use the chain rule.
First, let's simplify the expression to make it easier to differentiate.
ln(sqrt((1-ln(x))/(1+ln(x)))) can be rewritten as 1/2 * ln((1-ln(x))/(1+ln(x))).
Now, let's apply the chain rule:
1. Start by finding the derivative of the outer function, which is ln(u), where u = (1-ln(x))/(1+ln(x)). The derivative of ln(u) is 1/u times the derivative of u.
2. Next, find the derivative of the inner function, u = (1-ln(x))/(1+ln(x)). To differentiate this, we'll use the quotient rule.
a. Let f(x) = 1-ln(x) and g(x) = 1+ln(x).
b. Apply the quotient rule: (f'(x)g(x) - g'(x)f(x)) / g(x)^2.
c. Differentiate f(x) and g(x) step by step:
- Differentiate f(x): f'(x) = -1/x.
- Differentiate g(x): g'(x) = 1/x.
d. Plug these values into the quotient rule:
( (-1/x) * (1 + ln(x)) - (1/x) * (1 - ln(x)) ) / (1 + ln(x))^2
3. Multiply the result obtained in step 2 with the derivative of the outer function:
1/u * ( (-1/x) * (1 + ln(x)) - (1/x) * (1 - ln(x)) ) / (1 + ln(x))^2
4. Simplify the expression:
1/(2u) * ( (-1/x) * (1 + ln(x)) - (1/x) * (1 - ln(x)) ) / (1 + ln(x))^2
Now, combine like terms if necessary and simplify the expression further if possible. This revised expression is the derivative of ln(sqrt((1-ln(x))/(1+ln(x)))).