Differentiate the following w.r.t. x :
`x^(x^(2)-3)+(x-3)^(x^(2)),"for "xgt3`

Answer

To differentiate the function \( y = x^{(x^2 - 3)} + (x - 3)^{x^2} \) with respect to \( x \), we will use logarithmic differentiation and the product rule. Here’s a step-by-step solution:

### Step 1: Define the function
Let:
\[ y = x^{(x^2 - 3)} + (x - 3)^{x^2} \]

### Step 2: Differentiate using the sum rule
We can differentiate \( y \) using the sum rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(x^{(x^2 - 3)}) + \frac{d}{dx}((x - 3)^{x^2}) \]

### Step 3: Differentiate the first term \( u = x^{(x^2 - 3)} \)
To differentiate \( u = x^{(x^2 - 3)} \), we will use logarithmic differentiation:
1. Take the natural logarithm:
\[ \ln u = (x^2 - 3) \ln x \]
2. Differentiate both sides:
\[ \frac{1}{u} \frac{du}{dx} = \frac{d}{dx}((x^2 - 3) \ln x) \]
Using the product rule:
\[ \frac{d}{dx}((x^2 - 3) \ln x) = (2x) \ln x + (x^2 - 3) \frac{1}{x} \]
Therefore:
\[ \frac{du}{dx} = u \left( 2x \ln x + \frac{x^2 - 3}{x} \right) \]
Substituting back \( u = x^{(x^2 - 3)} \):
\[ \frac{du}{dx} = x^{(x^2 - 3)} \left( 2x \ln x + \frac{x^2 - 3}{x} \right) \]

### Step 4: Differentiate the second term \( v = (x - 3)^{x^2} \)
Similarly, for \( v = (x - 3)^{x^2} \):
1. Take the natural logarithm:
\[ \ln v = x^2 \ln(x - 3) \]
2. Differentiate both sides:
\[ \frac{1}{v} \frac{dv}{dx} = \frac{d}{dx}(x^2 \ln(x - 3)) \]
Again using the product rule:
\[ \frac{d}{dx}(x^2 \ln(x - 3)) = (2x) \ln(x - 3) + x^2 \frac{1}{x - 3} \]
Therefore:
\[ \frac{dv}{dx} = v \left( 2x \ln(x - 3) + \frac{x^2}{x - 3} \right) \]
Substituting back \( v = (x - 3)^{x^2} \):
\[ \frac{dv}{dx} = (x - 3)^{x^2} \left( 2x \ln(x - 3) + \frac{x^2}{x - 3} \right) \]

### Step 5: Combine the results
Now we can combine the derivatives:
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]
Substituting the expressions for \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = x^{(x^2 - 3)} \left( 2x \ln x + \frac{x^2 - 3}{x} \right) + (x - 3)^{x^2} \left( 2x \ln(x - 3) + \frac{x^2}{x - 3} \right) \]

### Final Answer
Thus, the derivative of the function \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = x^{(x^2 - 3)} \left( 2x \ln x + \frac{x^2 - 3}{x} \right) + (x - 3)^{x^2} \left( 2x \ln(x - 3) + \frac{x^2}{x - 3} \right) \]