Given:
y = log(tan(x))

To find:
dy/dx

Solution:

Step 1: Rewrite the given expression
As we know, the logarithm of a number to a given base is equal to the logarithm of the number divided by the logarithm of the base. Therefore, we can rewrite the given expression as:

y = log(tan(x)) = log(tan(x))/log(10)

Step 2: Take the derivative
Now, let's find the derivative of y with respect to x, which is dy/dx.

Using the quotient rule, the derivative of y can be expressed as:

dy/dx = (d/dx)[log(tan(x))/log(10)]

Step 3: Apply the chain rule
The derivative of a logarithmic function with respect to x can be calculated using the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x).

In this case, we have log(tan(x))/log(10) as the composite function. Let's break it down further:

Let u = log(tan(x))
Let v = log(10)

Therefore, y = u/v

Now, we can find the derivative of y using the chain rule:

dy/dx = (du/dx * v - u * dv/dx) / v^2

Step 4: Find the derivatives of u and v
To calculate the derivatives du/dx and dv/dx, we need to find the derivatives of u and v separately:

Derivative of u:
Using the chain rule, we have:

du/dx = d/dx[log(tan(x))] = (1/tan(x)) * (d/dx)[tan(x)]

The derivative of tan(x) is sec^2(x), so:

du/dx = (1/tan(x)) * sec^2(x)

Derivative of v:
The derivative of log(10) with respect to x is zero since it is a constant.

Therefore, dv/dx = 0

Step 5: Substitute the derivatives into the derivative formula
Now, let's substitute the derivatives we found into the derivative formula:

dy/dx = [(1/tan(x)) * sec^2(x) * log(10) - log(tan(x)) * 0] / (log(10))^2

Simplifying further:

dy/dx = (sec^2(x) * log(10)) / (tan(x) * log(10))^2

Final Answer:
Therefore, the derivative of y = log(tan(x)) with respect to x is:

dy/dx = (sec^2(x) * log(10)) / (tan(x) * log(10))^2