Question

Derivative, I can't get it. How can I solve it?

Answer 1

For starters, you can write


`(x√(x+1))/(x+1)=(x√(x+1))/((√(x+1))^2)=\frac x{√(x+1)}=\frac x{(x+1)^(1/2)}`

since the original function is only defined and continuous for
`x>-1`, so you never divide by 0.

Now to differentiate, you can use the quotient rule:


`(\mathrm d)/(\mathrm dx)\left[\frac x{(x+1)^(1/2)}\right]=((x+1)^(1/2)(\mathrm d)/(\mathrm dx)x-x(\mathrm d)/(\mathrm dx)\left[(x+1)^(1/2)\right])/(\left((x+1)^(1/2)\right)^2)=\frac{(x+1)^(1/2)-\frac x{2(x+1)^(1/2)}}{x+1}`

Next, factor out
`\frac1{2(x+1)^(1/2)}` to get


`(2(x+1)-x)/(2(x+1)^(1/2)(x+1))=(x+2)/(2(x+1)^(3/2))`