Using the quotient rule, differentiate $y = \dfrac{2x-3}{x^2+4}$.

Using the quotient rule, differentiate y = (2x-3)/(x²+4).

Worked Examples · Example 4

y = \dfrac{2x-3}{x^2+4}

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Given: $y=\dfrac{2x-3}{x^2+4}$ .

Step 1 — Identify $u$ and $v$ : $u=2x-3$ , $v=x^2+4$ , so $u'=2$ , $v'=2x$ .

Step 2 — Apply the quotient rule: $\dfrac{dy}{dx}=\dfrac{vu'-uv'}{v^2}=\dfrac{(x^2+4)(2)-(2x-3)(2x)}{(x^2+4)^2}$ .

Step 3 — Expand the numerator: $(x^2+4)(2)=2x^2+8$ ; $(2x-3)(2x)=4x^2-6x$ .

Step 4 — Subtract (careful with order): $2x^2+8-(4x^2-6x) = -2x^2+6x+8$ .

Step 5 — Write the final result: $\dfrac{dy}{dx}=\dfrac{-2x^2+6x+8}{(x^2+4)^2}$ . …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2024Set ANNUAL1 markMCQ
$5x^4 - 3x^3 + 2x^2 - 11x + 7$
(a) $20x^3 + 9x^2 - 4x + 11$
(b) $20x^3 - 9x^2 + 4x - 11$
(c) $20x^3 - 9x^2 - 4x - 11$
(d) $20x^3 + 9x^2 + 4x + 11$
›Reveal solutionSolution

Apply the power rule term by term: the derivative is $20x^3-9x^2+4x-11$ .

Using $\dfrac{d}{dx}(x^n)=nx^{n-1}$ on $5x^4-3x^3+2x^2-11x+7$ :

$\frac{d}{dx}(5x^4)=20x^3,\quad \frac{d}{dx}(-3x^3)=-9x^2,$ …

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CBSE 2024Set ANNUAL1 mark
$Q. Answer each of the following question in one sentence each : Write the formula for determining the derivative of the product of two functions.$
›Reveal solutionSolution

Product rule: $\dfrac{d}{dx}(uv)=u\,v'+v\,u'$ .

If $u=u(x)$ and $v=v(x)$ are two differentiable functions, the derivative of their product is

$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}.$ …

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CBSE 2023Set ANNUAL1 markMCQ
$\dfrac{d}{dx}(5x^6)$
(a) $x^{30}$
(b) $5x^5$
(c) $30x^5$
(d) $30x$
›Reveal solutionSolution

Using the power rule, $\dfrac{d}{dx}(5x^6)=30x^5$ .

Step 1 — the power rule states $\dfrac{d}{dx}(x^n)=nx^{n-1}$ .

Step 2 — constants multiply through (constant-multiple rule):

…

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CBSE 2019Set ANNUAL1 mark
$Q. Answer each of the following questions in one sentence:$
(xi) Differentiate $3r^2 + 4r + 7$ with respect to $r$ .
›Reveal solutionSolution

The derivative is $6r + 4$ .

Using the power rule $\dfrac{d}{dr}(r^n) = n r^{n-1}$ and the fact that the derivative of a constant is $0$ , differentiate each term: …

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CBSE 2019Set ANNUAL1 markMCQ
$\sqrt{x}$
(a) $\dfrac{1}{2}\sqrt{x}$
(b) $\dfrac{1}{2} x^{3/2}$
(c) $\dfrac{1}{2} x\sqrt{x}$
(d) $\dfrac{1}{2\sqrt{x}}$
›Reveal solutionSolution

Correct option: (d) $\dfrac{1}{2\sqrt{x}}$ .

Express the root as a power and use $\dfrac{d}{dx}x^n = n x^{n-1}$ : …

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←Previous questionExample 3 — Using the product rule, differentiate y = (3x+1)(x²-2).→Next questionExample 5 — Using the chain rule, differentiate y = (3x² - 5x)⁴.

Q.Using the quotient rule, differentiate $y = \dfrac{2x-3}{x^2+4}$ .

Concept understanding — Standard Rules of Differentiation

Q.Using the quotient rule, differentiate y=2x−3x2+4y = \dfrac{2x-3}{x^2+4}y=x2+42x−3​.

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Q.Using the quotient rule, differentiate $y = \dfrac{2x-3}{x^2+4}$ .

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