Question

If y=(1+x¼)(1+x½)(1-x¼), then dy/dx=...?​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\displaystyle\,y=\left(1+x^{\frac{1}{4}}\right)\cdot\left(1+x^{\frac{1}{2}}\right)\cdot\left(1-x^{\frac{1}{4}}\right)}$

$\tt{\displaystyle\,\implies\,y=\left(1+x^{\frac{1}{4}}\right)\cdot\left(1-x^{\frac{1}{4}}\right)\cdot\left(1+x^{\frac{1}{2}}\right)}$

$\tt{\displaystyle\,\implies\,y=\left\{(1)^2-\left(x^{\frac{1}{4}}\right)^2\right\}\cdot\left(1+x^{\frac{1}{2}}\right)}$

$\tt{\displaystyle\,\implies\,y=\left(1-x^{\frac{1}{2}}\right)\cdot\left(1+x^{\frac{1}{2}}\right)}$

$\tt{\displaystyle\,\implies\,y=\left\{(1)^2-\left(x^{\frac{1}{2}}\right)^2\right\}}$

$\tt{\displaystyle\,\implies\,y=1-x}$

$\tt{\displaystyle\,\implies\,\dfrac{dy}{dx}=-1}$

Answer 2

$y = (1 + \sqrt[4]{x} )(1 + \sqrt{x} )(1 - \sqrt[4]{x} )$

Further simplification of y, we get,

$y = \{ {(1)}^{2} - {( \sqrt[4]{x} )}^{2} \} \{1 + \sqrt{x} \} \\ y = (1 - \sqrt{x} )(1 + \sqrt{x} ) \\ y = \{ {(1)}^{2} - {( \sqrt{x} )}^{2} \} \\ y= 1 - x$

Now,

y = 1 - x

dy/dx = d(1 - x)/dx

= (0 - 1) = - 1

Hence,

dy/dx = -1