Question

How do you find the second derivative of `f(t)=tsqrtt`?

Answer 1

`(d^2f)/(dt^2)=3/(4sqrtt)`

Explanation:

As `tsqrtt` can be written as `f(t)=tsqrtt=txxt^(1/2)=t^(3/2)`, we can use the formula `d/(dx) x^n=nx^(n-1)`

Hence `(df)/(dt)=3/2xxt^(3/2-1)=3/2t^(1/2)=3/2sqrtt`

and `(d^2f)/(dt^2)=3/2xx1/2xxt^(1/2-1)=3/4t^(-1/2)=3/(4sqrtt)`

Answer 2

`f''(t)=3/(4sqrtt)`

Explanation:

`f(t)=tsqrtt=txxt^(1/2)=t^(3/2)`

`"differentiate using the "color(blue)"power rule"`

`• d/dx(ax^n)=nax^(n-1)`

`rArrf'(t)=3/2t^(1/2)`

`rArrf''(t)=3/2xx1/2t^(-1/2)`

`color(white)(rArrf''(t))=3/4xx1/t^(1/2)=3/(4sqrtt)`