Q.Differentiate $\sqrt{\dfrac{(x-3)(x^2+4)}{3x^2+4x+5}}$ w.r.t. $x$.

Concept: Implicit Differentiation — but here we use logarithmic differentiation because the function is a complicated product/quotient under a square root.

Let

Take natural logs on both sides:

Differentiate w.r.t. $x$:

Multiply both sides by $y$:

When you see a function that is a product or quotient of several expressions, all inside a square root, the usual quotient rule and product rule would be a nightmare. There’s a cleaner way: logarithmic differentiation.

The idea is simple. Instead of differentiating $y = f(x)$ directly, we take the natural log of both sides, use log properties to break the expression into a sum of simpler terms, and then differentiate implicitly. The logarithm turns multiplication into addition, division into subtraction, and powers into coefficients. That makes the derivative much easier to handle.

Let’s apply it here.

1. Set up the function. Let

We can also write this as

2. Take natural logs on both sides.

3. Use log properties to expand. The log of a quotient is the difference of logs, and the log of a product is the sum:

4. Differentiate both sides with respect to $x$. On the left, by the chain rule:

On the right, differentiate term by term:

So we have:

5. Solve for $\frac{dy}{dx}$. Multiply both sides by $y$:

6. Substitute back $y$. Remember $y = \sqrt{\frac{(x-3)(x^2+4)}{3x^2+4x+5}}$. So:

7. Optional: combine into a single fraction (if needed). For most exam purposes, the expression above is perfectly acceptable. But if you want a single rational expression, you can combine the three terms inside the parentheses over a common denominator. That’s just algebraic cleanup — the calculus is done.

Method: Logarithmic Differentiation for Products, Quotients, and Roots

Use this method whenever the function to differentiate is a product, quotient, or root of several factors combined together — situations where directly applying the product/quotient rule would be long and error-prone.

Steps

Step 1: Take the natural logarithm of both sides

If $y$ = (a product/quotient/root of several factors), write $\log y = \log(\dots)$.

Step 2: Use logarithm laws to break the right side into a sum of simple terms

A square root is a power of $\tfrac12$, so it contributes an overall factor of $\tfrac12$ in front of the whole expanded sum.

Step 3: Differentiate both sides with respect to $x$

The left side becomes $\dfrac{1}{y}\dfrac{dy}{dx}$ (chain rule); the right side becomes a sum of simple terms of the form $\dfrac{g'(x)}{g(x)}$, one per factor.

Step 4: Multiply both sides by $y$ and substitute the original expression back in

Applying to this type of problem: the more factors multiplied, divided, or nested inside a root, the more this method pays off — each factor contributes exactly one simple additive term after taking logs, rather than a tangle of nested product/quotient rules.

Common Mistakes

Mistake 1: Dropping the overall factor of $\frac12$ from the square root

Why it's wrong: Since $y$ is a square root, i.e. a power of $\tfrac12$, taking logs brings that $\tfrac12$ out in front of the entire expanded sum of logs — forgetting it means every term in the final derivative is exactly double what it should be. Correct approach: write $\log y = \tfrac12[\dots]$ right from the start, before expanding the individual log terms.

Mistake 2: Getting the sign wrong on the log of the denominator

Why it's wrong: The quotient rule for logs gives $\log\left(\frac{A}{B}\right) = \log A - \log B$ — the denominator's log term is subtracted, not added. Missing this sign flip turns a subtraction into an addition in the final derivative. Correct approach: explicitly write out $-\log(3x^2+4x+5)$ as a separate, subtracted term before differentiating.