Question

Find the Derived Function

Answer 1

a)
`(dy)/(dx) = (1-3lnx)/(x^4)`

b)
`(dy)/(dx) = \frac{-2}{3\sqrt[3]{1-x} }`

Step-by-step explanation:

a)
`y=(lnx)/(x^3)`

`y = \frac{lnx}{\sqrt[3]{x} }`

`y = lnx. x^-3`

Differentiating the above equation in terms of x

`(dy)/(dx) = (1)/(x) * x^-3 - 3lnx* x^-^4\\(dy)/(dx) = (1)/(x^4) - (3lnx)/(x^4) \\(dy)/(dx) = (1-3lnx)/(x^4) \\`

b)
`y = \sqrt[3]{1-x^(2) }`

Differentiating the above equation in terms of x

`y = \sqrt[3]{(1-x)^(2) } \\(dy)/(dx) = (1-x)^(2)/(3) \\(dy)/(dx) = (2)/(3)* (1-x)^(-1)/(3) * -1\\(dy)/(dx) = \frac{-2}{3\sqrt[3]{1-x} }`

Thus,

a)
`(dy)/(dx) = (1-3lnx)/(x^4)`

b)
`(dy)/(dx) = \frac{-2}{3\sqrt[3]{1-x} }`