Differentiate the following w.r.t.x :
`cosx^(3).sin^(2)(x^(5))`

Answer

To differentiate the function \( y = \cos(x^3) \cdot \sin^2(x^5) \) with respect to \( x \), we will apply the product rule and the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the Functions
We have two functions:
- \( u = \cos(x^3) \)
- \( v = \sin^2(x^5) \)

### Step 2: Apply the Product Rule
The product rule states that if \( y = u \cdot v \), then:
\[ \frac{dy}{dx} = u'v + uv' \]

### Step 3: Differentiate \( u \)
To differentiate \( u = \cos(x^3) \), we use the chain rule:
\[ u' = -\sin(x^3) \cdot \frac{d}{dx}(x^3) = -\sin(x^3) \cdot 3x^2 \]

### Step 4: Differentiate \( v \)
To differentiate \( v = \sin^2(x^5) \), we again use the chain rule:
\[ v' = 2\sin(x^5) \cdot \frac{d}{dx}(\sin(x^5)) = 2\sin(x^5) \cdot \cos(x^5) \cdot \frac{d}{dx}(x^5) = 2\sin(x^5) \cdot \cos(x^5) \cdot 5x^4 \]
This simplifies to:
\[ v' = 10x^4 \sin(x^5) \cos(x^5) \]

### Step 5: Substitute Back into the Product Rule
Now we substitute \( u', v, u, \) and \( v' \) back into the product rule:
\[ \frac{dy}{dx} = (-3x^2 \sin(x^3)) \cdot \sin^2(x^5) + \cos(x^3) \cdot (10x^4 \sin(x^5) \cos(x^5)) \]

### Step 6: Simplify the Expression
We can rearrange and simplify the expression:
\[ \frac{dy}{dx} = -3x^2 \sin(x^3) \sin^2(x^5) + 10x^4 \cos(x^3) \sin(x^5) \cos(x^5) \]

### Final Answer
Thus, the derivative of \( y = \cos(x^3) \cdot \sin^2(x^5) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -3x^2 \sin(x^3) \sin^2(x^5) + 10x^4 \cos(x^3) \sin(x^5) \cos(x^5) \]