Q.Find $\frac{dy}{dx}$ in the following: $x^3 \log x$

Concept: Chain Rule — differentiate the outer function (cube) first, then multiply by the derivative of the inner function ($\log x$).

Let $y = x^3 \log x$. This is a product, not a composition, so we use the product rule:

Step 1: $\frac{d}{dx}(x^3) = 3x^2$

Step 2: $\frac{d}{dx}(\log x) = \frac{1}{x}$ (assuming base $e$)

Substitute: …

The function given is $y = x^3 \log x$. This is a product of two distinct functions: $x^3$ (a power function) and $\log x$ (the natural logarithm). Whenever you have a product of two functions, the natural tool is the product rule, not the chain rule. The chain rule would apply if we had a composition like $\log(x^3)$ or $(x^3)^2$, but here the functions are multiplied, not nested.

The product rule states: if $y = u \cdot v$, then

Let’s apply it step by step.

1. Identify the two factors.

   Let $u = x^3$ and $v = \log x$.

   (Here $\log x$ means the natural logarithm, base $e$, as is standard in calculus.)

2. Differentiate each factor separately.

   • Derivative of $u = x^3$:

• Derivative of $v = \log x$:

3. Apply the product rule.

4. Simplify the first term. $x^3 \cdot \frac{1}{x} = x^2$. So we have:

5. Factor if desired (optional). Both terms share a factor $x^2$, so we can write: …

Method: The Product Rule (with Chain Rule on Each Factor)

When two functions of $x$ are multiplied together, neither the sum rule nor differentiating each factor separately and multiplying works — the product rule is required.

Steps

Step 1: Identify the two factors $u(x)$ and $v(x)$ being multiplied

Step 2: Differentiate each factor separately

If either factor is itself composite, apply the chain rule to it individually at this stage.

Step 3: Combine using the product rule

$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}.$ …

Common Mistakes

Mistake 1: Treating $\log x$ as a constant and only differentiating $x^3$.

Why it's wrong: $\log x$ is itself a function of $x$ and contributes its own term via the product rule — skipping it loses the entire $3x^2\log x$ piece of the answer. Correct approach: identify both factors as functions of $x$ before applying the product rule.

Mistake 2: Forgetting to simplify $x^3\cdot\frac{1}{x}$ to $x^2$. …