`y=10^(10^(x))`, find `(dy)/(dx)`

Answer

To find the derivative \(\frac{dy}{dx}\) for the function \(y = 10^{10^x}\), we can follow these steps:

### Step 1: Rewrite the Function
We start with the given function:
\[ y = 10^{10^x} \]

### Step 2: Apply the Chain Rule
To differentiate \(y\) with respect to \(x\), we use the chain rule. The derivative of \(a^u\) with respect to \(x\) is given by:
\[ \frac{d}{dx}(a^u) = a^u \cdot \ln(a) \cdot \frac{du}{dx} \]
Here, \(a = 10\) and \(u = 10^x\).

### Step 3: Differentiate the Outer Function
First, we differentiate the outer function:
\[ \frac{dy}{dx} = 10^{10^x} \cdot \ln(10) \cdot \frac{d}{dx}(10^x) \]

### Step 4: Differentiate the Inner Function
Next, we need to differentiate the inner function \(10^x\):
\[ \frac{d}{dx}(10^x) = 10^x \cdot \ln(10) \]

### Step 5: Substitute Back
Now, we substitute this back into our expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = 10^{10^x} \cdot \ln(10) \cdot (10^x \cdot \ln(10)) \]

### Step 6: Simplify the Expression
We can simplify this expression:
\[ \frac{dy}{dx} = 10^{10^x} \cdot 10^x \cdot (\ln(10))^2 \]
This can be written as:
\[ \frac{dy}{dx} = 10^{10^x + x} \cdot (\ln(10))^2 \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 10^{10^x + x} \cdot (\ln(10))^2 \]

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