Q.Differentiate with respect to $x$: $\dfrac{3x + 4}{5x^2 - 7x + 9}$.

To differentiate the given rational function, we use the Quotient Rule.

Let $u = 3x + 4$ and $v = 5x^2 - 7x + 9$.

Then, $\dfrac{du}{dx} = 3$ and $\dfrac{dv}{dx} = 10x - 7$.

Applying the Quotient Rule:

Expanding the numerator:

$= \dfrac{(15x^2 - 21x + 27) - (30x^2 - 21x + 40x - 28)}{(5x^2 - 7x + 9)^2}$ …

The function we need to differentiate, $\dfrac{3x + 4}{5x^2 - 7x + 9}$, is a ratio of two distinct functions of $x$. When we encounter a function that is expressed as a fraction, where both the numerator and the denominator are functions of the variable we are differentiating with respect to, the most direct and appropriate method to find its derivative is the Quotient Rule.

The Quotient Rule essentially tells us how the rate of change of a ratio behaves. It accounts for how changes in the numerator and denominator individually contribute to the overall change in the fraction, while also considering the "weight" of the denominator. Imagine a fraction $\frac{u}{v}$. If $u$ increases, the fraction increases. If $v$ increases, the fraction decreases. The Quotient Rule combines these effects precisely.

Let's apply this rule step-by-step.

1. Identify the numerator and denominator functions.

   We define $u(x)$ as the numerator and $v(x)$ as the denominator:

   $u(x) = 3x + 4$

   $v(x) = 5x^2 - 7x + 9$

2. Differentiate $u(x)$ and $v(x)$ with respect to $x$.

   We use the power rule and the sum/difference rule for differentiation.

   For $u(x) = 3x + 4$:

   $\dfrac{du}{dx} = \dfrac{d}{dx}(3x) + \dfrac{d}{dx}(4) = 3 \cdot x^{1-1} + 0 = 3 \cdot 1 + 0 = 3$

   For $v(x) = 5x^2 - 7x + 9$:

   $\dfrac{dv}{dx} = \dfrac{d}{dx}(5x^2) - \dfrac{d}{dx}(7x) + \dfrac{d}{dx}(9)$

   $\dfrac{dv}{dx} = 5 \cdot (2x^{2-1}) - 7 \cdot x^{1-1} + 0 = 5 \cdot (2x) - 7 \cdot 1 + 0 = 10x - 7$

   So, we have:

   $u(x) = 3x + 4$

   $u'(x) = 3$

   $v(x) = 5x^2 - 7x + 9$

   $v'(x) = 10x - 7$

3. Apply the Quotient Rule formula.

   Substitute the identified functions and their derivatives into the Quotient Rule formula:

> [!WARNING]
> A common mistake is to swap the order of terms in the numerator, i.e., $u v' - v u'$. Remember that the term with $v u'$ comes first, and the subtraction is crucial. The denominator is always $v^2$.

4. Simplify the numerator. …