Question

Given the function y=-\frac{3\sqrt{x^5}}{4},y=− 4 3 x 5 ​ ​ , find \frac{dy}{dx}. dx dy ​ . Express your answer in radical form without using negative exponents, simplifying all fractions.

Answer 1

Given:

The function is:

`y=-(3√(x^5))/(4)`

To find:

The value of
`(dy)/(dx)`.

Solution:

We have,

`y=-(3√(x^5))/(4)`

It can be written as

`y=-(3)/(4)x^{(5)/(2)}`
`[\because \sqrt[n]x=x^(1)/(n)]`

Differentiate with respect to x.

`(dy)/(dx)=-(3)/(4)* (5)/(2)x^{(5)/(2)-1}`
`[\because (d)/(dx)x^n=nx^(n-1)]`

`(dy)/(dx)=-(15)/(8)x^{(5-2)/(2)}`

`(dy)/(dx)=-(15)/(8)x^{(3)/(2)}`

`(dy)/(dx)=-(15)/(8)√(x^3)`

Therefore, the value of
`(dy)/(dx)` is
`-(15)/(8)√(x^3)`.