Q17(iii):

Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (iii) by logarithmic differentiation. Do they all give the same answer?

Step 1: Establishing the Logarithmic Framework

Given the function $y = (x^2 - 5x + 8)(x^3 + 7x + 9)$, we apply the natural logarithm to both sides to simplify the product into a sum of logarithms.

$\ln(y) = \ln((x^2 - 5x + 8)(x^3 + 7x + 9))$

[By the property of logarithms: $\ln(ab) = \ln(a) + \ln(b)$]

$\ln(y) = \ln(x^2 - 5x + 8) + \ln(x^3 + 7x + 9)$

Step 2: Differentiating with respect to $x$

Differentiate both sides with respect to $x$ using the chain rule for the logarithmic terms, $\frac{d}{dx}(\ln(u)) = \frac{1}{u} \cdot \frac{du}{dx}$.

$\frac{d}{dx}(\ln(y)) = \frac{d}{dx}(\ln(x^2 - 5x + 8)) + \frac{d}{dx}(\ln(x^3 + 7x + 9))$

$\frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{x^2 - 5x + 8} \cdot \frac{d}{dx}(x^2 - 5x + 8) + \frac{1}{x^3 + 7x + 9} \cdot \frac{d}{dx}(x^3 + 7x + 9)$

Step 3: Evaluating Derivatives and Isolating $\frac{dy}{dx}$

Calculate the inner derivatives:

$\frac{d}{dx}(x^2 - 5x + 8) = 2x - 5$

$\frac{d}{dx}(x^3 + 7x + 9) = 3x^2 + 7$

Substitute these into the equation:

$\frac{1}{y} \frac{dy}{dx} = \frac{2x - 5}{x^2 - 5x + 8} + \frac{3x^2 + 7}{x^3 + 7x + 9}$

Isolate $\frac{dy}{dx}$ by multiplying by $y$:

$\frac{dy}{dx} = (x^2 - 5x + 8)(x^3 + 7x + 9) \left[ \frac{2x - 5}{x^2 - 5x + 8} + \frac{3x^2 + 7}{x^3 + 7x + 9} \right]$

Step 4: Final Algebraic Expansion

Multiply $y$ back into the brackets to reach the standard polynomial form:

$\frac{dy}{dx} = (2x - 5)(x^3 + 7x + 9) + (3x^2 + 7)(x^2 - 5x + 8)$

Expanding the terms:

$(2x^4 + 14x^2 + 18x - 5x^3 - 35x - 45) + (3x^4 - 15x^3 + 24x^2 + 7x^2 - 35x + 56)$

Grouping like terms:

$\frac{dy}{dx} = (2x^4 + 3x^4) + (-5x^3 - 15x^3) + (14x^2 + 24x^2 + 7x^2) + (18x - 35x - 35x) + (-45 + 56)$

$\frac{dy}{dx} = 5x^4 - 20x^3 + 45x^2 - 52x + 11$

Step 5: Verification of Multi-Method Consistency

Whether using the Product Rule, direct polynomial expansion, or logarithmic differentiation, the fundamental laws of calculus dictate that the derivative of a differentiable function is unique [Uniqueness Theorem of Derivatives].

Final Solution: Yes, all three methods yield the same result: $\frac{dy}{dx} = 5x^4 - 20x^3 + 45x^2 - 52x + 11$