Q.Differentiate w.r.t. $x$: $(x + 1)^2 (x + 2)^3 (x + 3)^4$.

Concept: Derivative Evaluation — Use logarithmic differentiation to handle a product of powers.

Let $y = (x + 1)^2 (x + 2)^3 (x + 3)^4$. Take natural log on both sides:

Differentiate w.r.t. $x$:

Multiply by $y$: …

Why logarithmic differentiation?

When you have a product of several functions raised to powers — like $(x+1)^2 (x+2)^3 (x+3)^4$ — the product rule alone would be a nightmare. You’d need to apply it repeatedly, and the algebra would balloon into a mess of nested terms.

Logarithmic differentiation sidesteps this. The trick: take the natural log of both sides first. The log turns multiplication into addition and powers into coefficients. Then differentiate — the chain rule handles the rest. Finally, multiply back the original function to get the derivative.

It’s clean, systematic, and works every time for products, quotients, and powers.

Step-by-step

1. Set up the function and take logs.

Let

Take the natural logarithm of both sides:

Using $\log(ab) = \log a + \log b$ and $\log(a^n) = n \log a$:

2. Differentiate both sides with respect to $x$.

On the left, by the chain rule:

On the right, differentiate term by term:

So we have:

3. Solve for $\frac{dy}{dx}$.

Multiply both sides by $y$:

Now substitute back $y = (x+1)^2 (x+2)^3 (x+3)^4$:

4. Simplify (optional but tidy).

You can combine the terms inside the bracket over a common denominator, but it’s not necessary for most exam contexts. If you do:

Then the derivative becomes: …

Method: Logarithmic Differentiation for a Product of Several Power Factors

This method is the standard approach whenever you need to differentiate a product of three or more factors, each raised to its own power — direct repeated application of the product rule becomes unmanageable, but logarithms turn the whole product into a simple sum.

Steps

Step 1: Take the natural log of both sides

For $y = (x+1)^2(x+2)^3(x+3)^4$,

using $\log(ab)=\log a+\log b$ and $\log(a^n)=n\log a$.

Step 2: Differentiate term by term

Each term is of the form $k\log(x+c)$, whose derivative is $\dfrac{k}{x+c}$ by the chain rule:

Step 3: Multiply through by $y$ …

Common Mistakes

Mistake 1: Forgetting to multiply back by $y$

Why it's wrong: after differentiating $\log y$, you only have $\dfrac1y\dfrac{dy}{dx}$ — reporting $\dfrac{2}{x+1}+\dfrac{3}{x+2}+\dfrac{4}{x+3}$ alone as "the derivative" omits the entire original product. Correct approach: always multiply the bracket by $y=(x+1)^2(x+2)^3(x+3)^4$ as the last step.

Mistake 2: Misapplying $\log(a^n)=n\log a$ to the wrong factor's power

Why it's wrong: with three different exponents (2, 3, 4) attached to three different linear factors, it's easy to swap a coefficient onto the wrong term when writing out $\log y$, which silently corrupts every later step. Correct approach: write out $\log y = 2\log(x+1)+3\log(x+2)+4\log(x+3)$ carefully, matching each exponent to its own factor before differentiating. …