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

Answer

To find the derivative of the function \( y = \log(x^3 + 3) \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \log(x^3 + 3) \]
Let \( u = x^3 + 3 \). Thus, we can rewrite the equation as:
\[ y = \log(u) \]

### Step 2: Differentiate using the chain rule
To find \( \frac{dy}{dx} \), we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Find \( \frac{dy}{du} \)
The derivative of \( y = \log(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \frac{1}{u} \]

### Step 4: Find \( \frac{du}{dx} \)
Now, we differentiate \( u = x^3 + 3 \) with respect to \( x \):
\[ \frac{du}{dx} = 3x^2 \]

### Step 5: Substitute back into the chain rule
Now we substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) back into the chain rule:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot 3x^2 \]

### Step 6: Substitute \( u \) back
Now, replace \( u \) with \( x^3 + 3 \):
\[ \frac{dy}{dx} = \frac{3x^2}{x^3 + 3} \]

### Final Answer
Thus, the derivative of \( y = \log(x^3 + 3) \) is:
\[ \frac{dy}{dx} = \frac{3x^2}{x^3 + 3} \]

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