derivative of log tan x

Answer

To find the derivative of \( \log(\tan x) \), we will apply the chain rule and the derivatives of logarithmic and trigonometric functions. Here’s the step-by-step solution:

### Step 1: Identify the function
We need to find the derivative of \( y = \log(\tan x) \).

### Step 2: Apply the chain rule
Using the chain rule, the derivative of \( \log(u) \) where \( u = \tan x \) is given by:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} \]
Here, \( u = \tan x \).

### Step 3: Find \( \frac{du}{dx} \)
The derivative of \( \tan x \) is:
\[ \frac{du}{dx} = \sec^2 x \]

### Step 4: Substitute back into the derivative formula
Now substituting \( u \) and \( \frac{du}{dx} \) back into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{\tan x} \cdot \sec^2 x \]

### Step 5: Simplify the expression
We can express \( \sec^2 x \) in terms of sine and cosine:
\[ \sec^2 x = \frac{1}{\cos^2 x} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{\sec^2 x}{\tan x} = \frac{1/\cos^2 x}{\sin x / \cos x} = \frac{1}{\cos^2 x} \cdot \frac{\cos x}{\sin x} = \frac{1}{\cos x \sin x} \]

### Step 6: Final result
The derivative of \( \log(\tan x) \) is:
\[ \frac{dy}{dx} = \frac{1}{\cos x \sin x} \]