If `y =e ^(x sin (x ^(3)))+(tan x)^(x) ` then `(dy)/(dx)` may be equal to:

Options

A. `e ^(x sin (x^(3)))[3x ^(3) cos (x ^(3))+sin (x ^(3)) ]+ (tan x) ^(x) [ln tan x+ 2x cosec 2x ]`
B. `e ^(x sin (x ^(3)))[x ^(3) cos (x ^(3))+ sin (x ^(3))]+ (tan x )^(x) [ln tan x+ 2x cosec 2x]`
C. `e ^(x sin ( x ^(3)))[x ^(3) sin (x ^(3))+sin (x ^(3))]+(tan x ) ^(x) [ln tan x+2 cosec 2x]`
D. `e ^(xsin (x ^(3)))[3x ^(3) cos (x ^(3))+sin (x ^(3))]+(tan x )^(x) [ln tan x+ (x sec ^(2))/(tan x )]`

Answer

To find the derivative of the function \( y = e^{x \sin(x^3)} + (\tan x)^x \), we will differentiate each term separately and then combine the results.

### Step-by-Step Solution:

1. **Differentiate the first term \( e^{x \sin(x^3)} \)**:
- We will use the chain rule for differentiation. Let \( u = x \sin(x^3) \).
- The derivative of \( e^u \) is \( e^u \cdot \frac{du}{dx} \).
- Now, we need to find \( \frac{du}{dx} \):
\[ u = x \sin(x^3) \]
- Using the product rule:
\[ \frac{du}{dx} = \sin(x^3) + x \cdot \frac{d}{dx}(\sin(x^3)) \]
- Now, differentiate \( \sin(x^3) \) using the chain rule:
\[ \frac{d}{dx}(\sin(x^3)) = \cos(x^3) \cdot 3x^2 \]
- Thus,
\[ \frac{du}{dx} = \sin(x^3) + x \cdot (3x^2 \cos(x^3)) = \sin(x^3) + 3x^3 \cos(x^3) \]
- Therefore, the derivative of the first term is:
\[ \frac{d}{dx}(e^{x \sin(x^3)}) = e^{x \sin(x^3)} \cdot \left(\sin(x^3) + 3x^3 \cos(x^3)\right) \]

2. **Differentiate the second term \( (\tan x)^x \)**:
- We will use logarithmic differentiation. Let \( v = (\tan x)^x \).
- Taking the natural logarithm:
\[ \ln v = x \ln(\tan x) \]
- Differentiating both sides:
\[ \frac{1}{v} \frac{dv}{dx} = \ln(\tan x) + x \cdot \frac{d}{dx}(\ln(\tan x)) \]
- Now, differentiate \( \ln(\tan x) \):
\[ \frac{d}{dx}(\ln(\tan x)) = \frac{1}{\tan x} \cdot \sec^2 x = \frac{\sec^2 x}{\tan x} \]
- Thus,
\[ \frac{dv}{dx} = v \left(\ln(\tan x) + x \cdot \frac{\sec^2 x}{\tan x}\right) \]
- Substituting back \( v = (\tan x)^x \):
\[ \frac{dv}{dx} = (\tan x)^x \left(\ln(\tan x) + x \cdot \frac{\sec^2 x}{\tan x}\right) \]

3. **Combine the derivatives**:
- The overall derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = e^{x \sin(x^3)} \left(\sin(x^3) + 3x^3 \cos(x^3)\right) + (\tan x)^x \left(\ln(\tan x) + x \cdot \frac{\sec^2 x}{\tan x}\right) \]

### Final Result:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = e^{x \sin(x^3)} \left(\sin(x^3) + 3x^3 \cos(x^3)\right) + (\tan x)^x \left(\ln(\tan x) + x \cdot \frac{\sec^2 x}{\tan x}\right) \]