Find the derivative of `y = 10^(3x)`

Answer

To find the derivative of the function \( y = 10^{3x} \), we can follow these steps:

### Step 1: Write down the function
We start with the function:
\[ y = 10^{3x} \]

### Step 2: Apply logarithmic differentiation
To differentiate \( y = 10^{3x} \), we can use logarithmic differentiation. We take the natural logarithm of both sides:
\[ \ln y = \ln(10^{3x}) \]

### Step 3: Simplify using logarithmic properties
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can simplify the right-hand side:
\[ \ln y = 3x \ln 10 \]

### Step 4: Differentiate both sides
Now, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(3x \ln 10) \]

Using the chain rule on the left side, we have:
\[ \frac{1}{y} \frac{dy}{dx} = 3 \ln 10 \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \cdot 3 \ln 10 \]

### Step 6: Substitute back for \( y \)
Since \( y = 10^{3x} \), we substitute back:
\[ \frac{dy}{dx} = 10^{3x} \cdot 3 \ln 10 \]

### Final Answer
Thus, the derivative of \( y = 10^{3x} \) is:
\[ \frac{dy}{dx} = 3 \ln 10 \cdot 10^{3x} \]

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