What is the rate of change of `sqrt(x^(2)+16)` with respect to `x^(2)` at x=3?

Options

A. `1//5`
B. `1//10`
C. `1//20`
D. none of the abvoe

Answer

B

Explanation

Let `u=sqrt(x^(2)+16), v=x^(2)`
`rarr u^(2)=x^(2)+16, (dv)/(dx)=2x`
`rarr 2u (du)/(dx)=2x`
`rarr(du)/(dx)=(x)/sqrt(x^(2)+16)`
Now required rate of change =`(du)/(dv)`
`=(du)/(dx)xx(dx)/(dv)=(x)/sqrt(x^(2)+16)xx(1)/(2x)=(1)(2sqtr(x^(2)+16)`
`Now (du)/(dv)|_(x=3) =(1)/(2xx5)=(1)/(10)`