Find the derivative of the functionf(x)= (x-5xsqrt(x))/(sqrt(x))

Expert Answers
justaguide eNotes educator| Certified Educator

The function f(x)= `(x - 5*x*sqrt x)/sqrt x`

The derivative is determined using the quotient rule.

f'(x) = `[(x - 5*x*sqrt x)'*sqrt x - (x - 5*x*sqrt x)*(sqrt x)']/(sqrt x)^2`

=> `((1 - 5*(3/2)*sqrt x)*sqrt x - (1/2)*(x - 5*x*sqrt x)/sqrt x)/x`

=> `(sqrt x - (15/2)*x - (1/2)*sqrt x + (5/2)*x)/x`

=> `((1/2)*sqrt x - 5*x)/x`

=> `(1/2)/sqrt x - 5`

The derivative of `(x - 5*x*sqrt x)/sqrt x` is `1/(2*sqrt x) - 5`

rcmath eNotes educator| Certified Educator

Another way to look at this problem is to simplify the function first, then differentiate.

`f(x)=x/sqrt(x)-(5xsqrt(x))/sqrt(x) =>`

```f(x)=sqrt(x)-5x=x^(1/2)-5x =>`


Depending on the teacher requirement, the answer can be simplified even more by rationalizing the denominator.