Question

How do I differentiate the following equations with roots?

a.) `root(3)(t)/(t-9`
b.) H(u) = (u-`sqrt(u)`) *(u + `sqrt(u)`)

Answer 1

`f'(t)=(1/3t^(-2/3)-2t^(1/3))/(t-9)^2`
`H'(u)=2u-1`

Explanation:

a)

We want to find the derivative of

`f(t)=root(3)t/(t-9)=t^(1/3)/(t-9)`

Use the Quotient Rule, if `f(t)=(h(t))/g(t)`,

Then `f'(t)=(h'(t)g(t)-h(t)g'(t))/(h(t))^2`

By the Quotient Rule with `h(t)=t^(1/3)` and `g(t)=t-9`
and the Power Rule for differentiation

`f'(t)=((d/dx(t^(1/3)))(t-9)-t^(1/3)(d/dx(t-9)))/(t-9)^2`

`=(1/3t^(-2/3)(t-9)-t^(1/3))/(t-9)^2`

`=(1/3t^(1/3)-3t^(-2/3)-t^(1/3))/(t-9)^2`

`=(-3t^(-2/3)-2/3t^(1/3))/(t-9)^2`

`=(-9t^(-2/3)-2t^(1/3))/(3(t-9)^2)`

`=(-9-2t)/(3(t-9)^2t^(2/3))`

b)

We want to find the derivative of

`H(u)=(u-sqrt(u))(u+sqrt(u))`

We could use the Product Rule

But here it would be easier to rewrite as

`H(u)=(u-sqrt(u))(u+sqrt(u))=u^2-u`

Use the Power Rule , if `f(x)=x^n`

Then `f'(x)=nx^(n-1)`

By the Power Rule

`H'(u)=2u^(2-1)-u^(1-1)`

`=2u-1`