Using the chain rule, differentiate $y = (3x^2 - 5x)^4$.

Using the chain rule, differentiate y = (3x² - 5x) – Class 12 Business Mathematics And Statistics

Worked Examples · Example 5

y = (3x^2 - 5x)^4

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Given: $y=(3x^2-5x)^4$ .

Step 1 — Set up the chain rule: let $u=3x^2-5x$ , so $y=u^4$ .

Step 2 — Differentiate the outer function: $\dfrac{dy}{du}=4u^3$ .

Step 3 — Differentiate the inner function: $\dfrac{du}{dx}=6x-5$ .

Step 4 — Multiply (chain rule): $\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}=4u^3(6x-5)=4(3x^2-5x)^3(6x-5)$ . …

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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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Q.Using the chain rule, differentiate $y = (3x^2 - 5x)^4$ .

Concept understanding — Standard Rules of Differentiation

Q.Using the chain rule, differentiate y=(3x2−5x)4y = (3x^2 - 5x)^4y=(3x2−5x)4.

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Q.Using the chain rule, differentiate $y = (3x^2 - 5x)^4$ .

Unlock everything free for 14 days