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

Differentiate with respect to x: (3x + 4)/(5x² - 7 – Class 11 Mathematics

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Concept understanding — Rules of Differentiation

Computing every derivative directly from the limit definition ("from first principle") is correct but impractical for anything beyond the simplest functions. The five rules below, proved once from the definition, make differentiation a mechanical, algebraic process for any combination of known derivatives — no further limit computation is needed once they are established. Throughout, $u=u(x)$ , $v=v(x)$ are differentiable functions of $x$ , and $k$ is a constant.

Sum Rule (Theorem 10.2). $\dfrac{d}{dx}(u+v)=\dfrac{du}{dx}+\dfrac{dv}{dx}$ — differentiate term by term. Extends to any finite sum.

Product Rule (Theorem 10.3). $\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx}+v\dfrac{du}{dx}$ , i.e. $(uv)'=uv'+vu'$ — "first times derivative of second, plus second times derivative of first." Extends to three or more factors, e.g. $(uvw)'=u'vw+uv'w+uvw'$ : differentiate one factor at a time, summing over which factor is differentiated.

Quotient Rule (Theorem 10.4). For $v\ne0$ , $\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{v\,u'-u\,v'}{v^2}$ — "bottom times derivative of top, minus top times derivative of bottom, all over bottom squared."

Chain Rule (Theorem 10.5). If $y=f(u)$ and $u=g(x)$ are both differentiable, so $y=f(g(x))$ , then

\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} = f'(g(x))\,g'(x)

— differentiate the outer function $f$ with respect to its argument $u=g(x)$ (the inner function), then multiply by the derivative of the inner function. This is the single most-used rule in the chapter: every composite function (a power of an expression, $\sin$ of an expression, $e^{\text{expression}}$ , and so on) needs it.

Constant Multiple Rule (Theorem 10.6). $\dfrac{d}{dx}\big[k\,f(x)\big] = k\,f'(x)$ — a constant factor simply carries through differentiation unchanged. …

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We differentiate the given rational function using the Quotient Rule, which involves finding the derivatives of the numerator and denominator, and then combining them according to the rule. The final derivative is $\boxed{\dfrac{-15x^2 - 40x + 55}{(5x^2 - 7x + 9)^2}}$ .

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.

y = \dfrac{u(x)}{v(x)}

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

Identify the numerator and denominator functions.

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

$u(x) = 3x + 4$

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

$u(x) = 3x + 4$

$u'(x) = 3$

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

$v'(x) = 10x - 7$
Apply the Quotient Rule formula.

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

\dfrac{dy}{dx} = \dfrac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}

\dfrac{dy}{dx} = \dfrac{(5x^2 - 7x + 9)(3) - (3x + 4)(10x - 7)}{(5x^2 - 7x + 9)^2}

> [!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. …

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Q.Differentiate with respect to xxx: 3x+45x2−7x+9\dfrac{3x + 4}{5x^2 - 7x + 9}5x2−7x+93x+4​.

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

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