Find the derivative of w.r.to x
`20 ^(log (tan x ))`

Answer

To find the derivative of the function \( y = 20^{\log(\tan x)} \) with respect to \( x \), we will use the following differentiation formulas:

1. \( \frac{d}{dx} a^x = a^x \log(a) \)
2. \( \frac{d}{dx} \log(x) = \frac{1}{x} \)
3. \( \frac{d}{dx} \tan(x) = \sec^2(x) \)

Now, let's differentiate the given function step by step.

### Step 1: Identify the function
We have:
\[ y = 20^{\log(\tan x)} \]

### Step 2: Differentiate using the exponential rule
Using the first formula, we differentiate \( y \):
\[ \frac{dy}{dx} = 20^{\log(\tan x)} \log(20) \cdot \frac{d}{dx}(\log(\tan x)) \]

### Step 3: Differentiate \( \log(\tan x) \)
Now we need to differentiate \( \log(\tan x) \):
Using the second formula:
\[ \frac{d}{dx}(\log(\tan x)) = \frac{1}{\tan x} \cdot \frac{d}{dx}(\tan x) \]
Now, using the third formula:
\[ \frac{d}{dx}(\tan x) = \sec^2(x) \]
So we have:
\[ \frac{d}{dx}(\log(\tan x)) = \frac{1}{\tan x} \cdot \sec^2(x) \]

### Step 4: Substitute back into the derivative
Now substituting back into our derivative:
\[ \frac{dy}{dx} = 20^{\log(\tan x)} \log(20) \cdot \left(\frac{\sec^2(x)}{\tan x}\right) \]

### Step 5: Simplify the expression
We can simplify this expression:
\[ \frac{dy}{dx} = 20^{\log(\tan x)} \cdot \log(20) \cdot \frac{\sec^2(x)}{\tan x} \]

### Final Answer
Thus, the derivative of \( y = 20^{\log(\tan x)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 20^{\log(\tan x)} \cdot \log(20) \cdot \frac{\sec^2(x)}{\tan x} \]