`d/(dx)(x^(4x^(3)))`=

Options

A. `x^(12x^(2))`
B. `4x^(4x^(3)+2).(1+3 log x)`
C. `4x^(3). Log x`
D. `4x^(3)(1+ 3 log x)`

Answer

To solve the differentiation problem \( \frac{d}{dx} (x^{4x^3}) \), we will follow these steps:

### Step 1: Rewrite the Function
Let \( y = x^{4x^3} \).

### Step 2: Take the Natural Logarithm
Taking the natural logarithm of both sides:
\[ \ln y = \ln (x^{4x^3}) = 4x^3 \ln x \]

### Step 3: Differentiate Both Sides
Now, differentiate both sides with respect to \( x \):
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} (4x^3 \ln x) \]

### Step 4: Apply the Product Rule
Using the product rule on the right side:
\[ \frac{d}{dx} (4x^3 \ln x) = 4x^3 \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}(4x^3) \]
\[ = 4x^3 \cdot \frac{1}{x} + \ln x \cdot (12x^2) \]
\[ = 4x^2 + 12x^2 \ln x \]

### Step 5: Substitute Back
Now substitute back into the equation:
\[ \frac{1}{y} \frac{dy}{dx} = 4x^2 + 12x^2 \ln x \]

### Step 6: Multiply Both Sides by \( y \)
Now, multiply both sides by \( y \):
\[ \frac{dy}{dx} = y (4x^2 + 12x^2 \ln x) \]

### Step 7: Substitute \( y \) Back
Substituting \( y = x^{4x^3} \):
\[ \frac{dy}{dx} = x^{4x^3} (4x^2 + 12x^2 \ln x) \]

### Final Answer
Thus, the derivative is:
\[ \frac{d}{dx} (x^{4x^3}) = x^{4x^3} (4x^2 + 12x^2 \ln x) \]

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