Find `dy/dx if y= 3^x`

Answer

To find \(\frac{dy}{dx}\) when \(y = 3^x\), we can follow these steps:

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

### Step 2: Take the natural logarithm of both sides
To differentiate \(y\), we take the natural logarithm of both sides:
\[ \ln(y) = \ln(3^x) \]

### Step 3: Apply the logarithmic property
Using the property of logarithms that states \(\ln(a^b) = b \cdot \ln(a)\), we can simplify the right side:
\[ \ln(y) = x \cdot \ln(3) \]

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

### Step 5: Apply the chain rule on the left side
Using the chain rule on the left side, we have:
\[ \frac{1}{y} \cdot \frac{dy}{dx} = \ln(3) \cdot \frac{d}{dx}(x) \]
Since the derivative of \(x\) with respect to \(x\) is 1, we get:
\[ \frac{1}{y} \cdot \frac{dy}{dx} = \ln(3) \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = y \cdot \ln(3) \]

### Step 7: Substitute back the value of \(y\)
Finally, we substitute back the value of \(y\):
\[ \frac{dy}{dx} = 3^x \cdot \ln(3) \]

Thus, the final answer is:
\[ \frac{dy}{dx} = 3^x \ln(3) \]

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