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find dy/dx when y= ((x)^2 -1)^3 (2x – 1)/ sqrt((x-3) (4x-1))

find dy/dx when
y= ((x)^2 -1)^3 (2x – 1)/ sqrt((x-3) (4x-1))

Grade:10

1 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
6 years ago
Hello Student,
\\ \frac{d}{dx}\left(\frac{\left(2x-1\right)\left(x^2-1\right)^3}{\sqrt{\left(x-3\right)\left(4x-1\right)}}\right) \\ =\frac{\frac{d}{dx}\left(\left(2x-1\right)\left(x^2-1\right)^3\right)\sqrt{\left(x-3\right)\left(4x-1\right)}-\frac{d}{dx}\left(\sqrt{\left(x-3\right)\left(4x-1\right)}\right)\left(2x-1\right)\left(x^2-1\right)^3}{\sqrt{\left(x-3\right)\left(4x-1\right)}^2} \\ \\ \frac{d}{dx}\left(\left(2x-1\right)\left(x^2-1\right)^3\right) \\ =\frac{d}{dx}\left(2x-1\right)\left(x^2-1\right)^3+\frac{d}{dx}\left(\left(x^2-1\right)^3\right)\left(2x-1\right) \\ \frac{d}{dx}\left(\left(x^2-1\right)^3\right) \\ =3\left(x^2-1\right)^22x \\ =2\left(x^2-1\right)^3+6x\left(x^2-1\right)^2\left(2x-1\right)
\\ =\frac{2\left(x^2-1\right)^2\left(x\left(7x-3\right)-1\right)\sqrt{\left(x-3\right)\left(4x-1\right)}-\frac{8x-13}{2\sqrt{\left(x-3\right)\left(4x-1\right)}}\left(2x-1\right)\left(x^2-1\right)^3}{\sqrt{\left(x-3\right)\left(4x-1\right)}^2} \\ =\frac{\left(x^2-1\right)^2\left(x\left(x\left(6x\left(16x-63\right)+227\right)-18\right)+1\right)}{2\left(\left(x-3\right)\left(4x-1\right)\right)^{\frac{3}{2}}}
Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty

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