Q.Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below:

The key idea is the Chain Rule (for logarithmic differentiation) and the Product Rule. We verify consistency by comparing results from three methods.

Step 1: Product Rule

Let $u = x^2 - 5x + 8$, $v = x^3 + 7x + 9$. Then $u' = 2x - 5$, $v' = 3x^2 + 7$.

Step 2: Expand and differentiate

First expand:

$(x^2 - 5x + 8)(x^3 + 7x + 9) = x^5 - 5x^4 + 15x^3 - 26x^2 + 11x + 72$

Differentiate termwise:

Step 3: Logarithmic differentiation

Take $\log y = \log(x^2 - 5x + 8) + \log(x^3 + 7x + 9)$. Differentiate: …

We are differentiating a product of two polynomials. The core idea is that the derivative of a product $u \cdot v$ is not simply $u' \cdot v'$ — that would be a common mistake. Instead, the product rule tells us: each piece gets a turn to be differentiated while the other stays unchanged, and we add the results. This is the heart of why the rule works: it accounts for how small changes in both factors contribute to the overall change.

Let’s work through each method step by step.

1. Using the product rule

Let $u = x^2 - 5x + 8$ and $v = x^3 + 7x + 9$.

First, find the derivatives:

• $u' = 2x - 5$
• $v' = 3x^2 + 7$

The product rule states: $\frac{d}{dx}(u v) = u' v + u v'$.

So:

Now expand each term carefully.

First term: $(2x - 5)(x^3 + 7x + 9)$

• $2x \cdot x^3 = 2x^4$
• $2x \cdot 7x = 14x^2$
• $2x \cdot 9 = 18x$
• $-5 \cdot x^3 = -5x^3$
• $-5 \cdot 7x = -35x$
• $-5 \cdot 9 = -45$

So first term = $2x^4 - 5x^3 + 14x^2 + (18x - 35x) - 45 = 2x^4 - 5x^3 + 14x^2 - 17x - 45$

Second term: $(x^2 - 5x + 8)(3x^2 + 7)$

• $x^2 \cdot 3x^2 = 3x^4$
• $x^2 \cdot 7 = 7x^2$
• $-5x \cdot 3x^2 = -15x^3$
• $-5x \cdot 7 = -35x$
• $8 \cdot 3x^2 = 24x^2$
• $8 \cdot 7 = 56$

So second term = $3x^4 - 15x^3 + (7x^2 + 24x^2) - 35x + 56 = 3x^4 - 15x^3 + 31x^2 - 35x + 56$

Now add them:

• $x^4$ terms: $2x^4 + 3x^4 = 5x^4$
• $x^3$ terms: $-5x^3 - 15x^3 = -20x^3$
• $x^2$ terms: $14x^2 + 31x^2 = 45x^2$
• $x$ terms: $-17x - 35x = -52x$
• Constant: $-45 + 56 = 11$

Thus:

2. By expanding the product first

Multiply the two polynomials directly:

$(x^2 - 5x + 8)(x^3 + 7x + 9)$

Multiply each term of the first by each term of the second:

• $x^2 \cdot x^3 = x^5$
• $x^2 \cdot 7x = 7x^3$
• $x^2 \cdot 9 = 9x^2$
• $-5x \cdot x^3 = -5x^4$
• $-5x \cdot 7x = -35x^2$
• $-5x \cdot 9 = -45x$
• $8 \cdot x^3 = 8x^3$
• $8 \cdot 7x = 56x$
• $8 \cdot 9 = 72$

Now combine like terms:

• $x^5$: $1x^5$
• $x^4$: $-5x^4$
• $x^3$: $7x^3 + 8x^3 = 15x^3$
• $x^2$: $9x^2 - 35x^2 = -26x^2$
• $x$: $-45x + 56x = 11x$
• Constant: $72$

So the expanded polynomial is:

Now differentiate term by term:

• $\frac{d}{dx}(x^5) = 5x^4$
• $\frac{d}{dx}(-5x^4) = -20x^3$
• $\frac{d}{dx}(15x^3) = 45x^2$
• $\frac{d}{dx}(-26x^2) = -52x$
• $\frac{d}{dx}(11x) = 11$
• $\frac{d}{dx}(72) = 0$

So:

This matches exactly.

3. By logarithmic differentiation …

Method: Cross-Checking a Derivative Using Three Independent Techniques

When a question asks you to differentiate the same function multiple ways, the point isn't just computing three times — it's understanding which technique suits which situation, and using agreement between methods as a genuine accuracy check.

Steps

Step 1: Product rule (the direct route)

For $y = uv$, use $\dfrac{dy}{dx} = u'v+uv'$ directly. This always works and needs no preliminary algebra — the natural first choice for a product of two functions.

Step 2: Expand-then-differentiate (only works for polynomials)

Multiply the factors out into a single polynomial first, then differentiate term by term using the power rule. This avoids the product rule entirely, but only works when the factors are polynomials (or otherwise easy to multiply out) — it fails immediately for trig, log, or exponential factors.

Step 3: Logarithmic differentiation (works even where expansion doesn't)

Take $\log y = \log u + \log v$, differentiate to get $\dfrac{y'}{y} = \dfrac{u'}{u}+\dfrac{v'}{v}$, then multiply back by $y = uv$ and simplify — the $u$ in the numerator cancels the $u$ in the first fraction's denominator, so this reduces algebraically to the same expression as the product rule.

Step 4: Confirm all three answers match …

Common Mistakes

Mistake 1: Sign errors while expanding the two polynomials directly

Why it's wrong: with nine cross-terms to multiply and combine, it's easy to drop a minus sign (especially from terms like $-5x\cdot 7x = -35x^2$) or miscombine like powers — a single sign slip changes the coefficient of one term in the final quartic without making the answer look obviously wrong. Correct approach: expand systematically term-by-term, and cross-check the result against the product-rule answer rather than trusting either method in isolation.

Mistake 2: In the logarithmic-differentiation method, forgetting to substitute $y=uv$ back in before finishing …