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Monotonically Increasing Functions

If y=2x2+6x2x...

Question

If $y = (2 x^{2} + 6 x) (2 x^{3} + 5 x^{2}),$ then $\frac{d y}{d x} =$

A

20 x^{4} + 80 x^{3} + 90 x^{2}

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B

20 x^{4} + 88 x^{3} + 90 x^{2}

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C

16 x^{4} + 88 x^{3} + 90 x^{2}

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D

16 x^{4} + 80 x^{3} + 90 x^{2}

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Solution

The correct option is B $20 x^{4} + 88 x^{3} + 90 x^{2}$
We have,
$y = (2 x^{2} + 6 x) (2 x^{3} + 5 x^{2})$
Let,
$u = 2 x^{2} + 6 x$ and $v = 2 x^{3} + 5 x^{2}$
Using Product Rule:
$\frac{d (u v)}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x}$
$\Rightarrow \frac{d u}{d x} = 4 x + 6$ and $\frac{d v}{d x} = 6 x^{2} + 10 x$
Therefore
$\frac{d y}{d x} = \frac{d (u v)}{d x} = u \frac{d v}{d x} +$ $v \frac{d u}{d x}$
$= (2 x^{2} + 6 x) (6 x^{2} + 10 x) + (2 x^{3} + 5 x^{2}) (4 x + 6)$
$= 20 x^{4} + 88 x^{3} + 90 x^{2}$

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Monotonically Increasing Functions

Standard XII Mathematics

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