ddx2x2−3x+52(5x−2)3=

ddx2x2−3x+52(5x−2)3=(5x−2)3⋅22x2−3x+5(4x−3)−2x2−3x+52⋅15(5x−2)2(5x−2)6=2x2−3x+5(5x−2)22(4x−3)(5x−2)−152x2−3x+5(5x−2)6=2x2−3x+510x2−x−63(5x−2)4.

ddx2x2−3x+52(5x−2)3=

ddx2x2−3x+52(5x−2)3=(5x−2)3⋅22x2−3x+5(4x−3)−2x2−3x+52⋅15(5x−2)2(5x−2)6=2x2−3x+5(5x−2)22(4x−3)(5x−2)−152x2−3x+5(5x−2)6=2x2−3x+510x2−x−63(5x−2)4.

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$\frac{d}{dx} \{\frac{{(2 x^{2} - 3 x + 5)}^{2}}{(5 x - 2)^{3}}\} =$

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$\frac{(2 x^{2} + 3 x + 5) (10 x^{2} - x - 63)}{(5 x - 2)^{4}}$

$\frac{(2 x^{2} - 3 x + 5) (10 x^{2} + x - 63)}{(5 x - 2)^{4}}$

$\frac{(2 x^{2} + 3 x + 5) (10 x^{2} + x - 63)}{(5 x - 2)^{4}}$

$\frac{(2 x^{2} - 3 x + 5) (10 x^{2} - x - 63)}{(5 x - 2)^{4}}$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{d}{dx} [\frac{{(2 x^{2} - 3 x + 5)}^{2}}{(5 x - 2)^{3}}] = \frac{(5 x - 2)^{3} \cdot 2 (2 x^{2} - 3 x + 5) (4 x - 3) - {(2 x^{2} - 3 x + 5)}^{2} \cdot 15 (5 x - 2)^{2}}{(5 x - 2)^{6}} \\ = \frac{(2 x^{2} - 3 x + 5) (5 x - 2)^{2} [2 (4 x - 3) (5 x - 2) - 15 (2 x^{2} - 3 x + 5)}{(5 x - 2)^{6}} \\ = \frac{(2 x^{2} - 3 x + 5) [10 x^{2} - x - 63]}{(5 x - 2)^{4}} . \end{array}$