Find the derivative of (ln(tan(x))/3pi
f= ln u
u= tan x
f'= 1/u *u'
u'= sec^2x x'
so
g'(x)=1/3PI * 1/tanx * sec^2 x checkall this.
To find the derivative of the function f(x) = (ln(tan(x)))/(3π), we can apply the quotient rule:
The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative is given by:
f'(x) = (g'(x)⋅h(x) - g(x)⋅h'(x))/(h(x))^2
In this case, g(x) = ln(tan(x)) and h(x) = 3π.
Let's find the derivatives of g(x) and h(x) first:
g'(x) = d/dx (ln(tan(x)))
= (1/tan(x))⋅sec^2(x)
= sec^2(x)/tan(x)
= sec(x)⋅csc(x)
h'(x) = d/dx (3π)
= 0
Now we can substitute these derivatives into the quotient rule formula:
f'(x) = (g'(x)⋅h(x) - g(x)⋅h'(x))/(h(x))^2
= (sec(x)⋅csc(x)⋅(3π) - ln(tan(x))⋅0)/(3π)^2
= (3π⋅sec(x)⋅csc(x))/(9π^2)
= sec(x)⋅csc(x)/(3π)
Therefore, the derivative of f(x) = (ln(tan(x)))/(3π) is f'(x) = sec(x)⋅csc(x)/(3π).
To find the derivative of the function f(x) = ln(tan(x))/(3π), we will use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x)/h(x), the derivative of f(x) can be found using the formula:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2
In our function f(x) = ln(tan(x))/(3π), g(x) = ln(tan(x)) and h(x) = 3π.
Now, let's find the derivatives of g(x) and h(x):
- The derivative of ln(tan(x)) can be found using the chain rule. Let u(x) = tan(x). Then, g(x) = ln(u) and g'(x) = (1/u) * u', where u' is the derivative of u. The derivative of tan(x) is sec^2(x), so g'(x) = (1/tan(x)) * sec^2(x).
- The derivative of h(x) = 3π is constant, so h'(x) = 0.
Now, we can substitute these values into the quotient rule formula:
f'(x) = [(1/tan(x)) * sec^2(x) * (3π) - ln(tan(x)) * 0] / [(3π)]^2
Simplifying further:
f'(x) = [(3π)/tan(x)] / [(9π^2)]
Now, let's simplify the expression:
f'(x) = (3π) / (9π^2 * tan(x))
Finally, we can simplify further by canceling out one of the π terms:
f'(x) = 1 / (3π * tan(x))
So, the derivative of f(x) = ln(tan(x))/(3π) is f'(x) = 1 / (3π * tan(x)).