Differentiate the following function with respect of `x :(1-2tanx)(5+4sinx)`

Explanation

To differentiate the function \( f(x) = (1 - 2\tan x)(5 + 4\sin x) \) with respect to \( x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by:

\[ \frac{d}{dx}(u \cdot v) = u'v + uv' \]

### Step-by-Step Solution:

1. **Identify the Functions**:
Let \( u = 1 - 2\tan x \) and \( v = 5 + 4\sin x \).

2. **Differentiate \( u \)**:
\[ u' = \frac{d}{dx}(1 - 2\tan x) = 0 - 2\sec^2 x = -2\sec^2 x \]

3. **Differentiate \( v \)**:
\[ v' = \frac{d}{dx}(5 + 4\sin x) = 0 + 4\cos x = 4\cos x \]

4. **Apply the Product Rule**:
Now, using the product rule:
\[ f'(x) = u'v + uv' \]
Substitute \( u \), \( u' \), \( v \), and \( v' \):
\[ f'(x) = (-2\sec^2 x)(5 + 4\sin x) + (1 - 2\tan x)(4\cos x) \]

5. **Simplify the Expression**:
Expanding both terms:
\[ f'(x) = -10\sec^2 x - 8\sin x \sec^2 x + 4\cos x - 8\tan x \cos x \]

6. **Combine Like Terms**:
We can rewrite \( -8\tan x \cos x \) as \( -8\sin x \):
\[ f'(x) = -10\sec^2 x - 8\sin x \sec^2 x + 4\cos x - 8\sin x \]

7. **Final Result**:
The derivative of the function is:
\[ f'(x) = -10\sec^2 x - 8\sin x \sec^2 x + 4\cos x - 8\sin x \]