Question

What is the derivative of `x^(3/2)log_2sqrt(x+1)`?

Answer 1

`(sqrtx)/(2ln2)[x/(x+1)+3*lnsqrt(x+1)]`

Explanation:

`Note:color(red)((1)log_ab=(log_eb)/(log_ea))`
`color(blue)((2)d/(dx)(u*v)=u*(dv)/(dx)+v*(du)/(dx))`
`y=x^(3/2)log_2sqrt(x+1)=x^(3/2)[(log_esqrt(x+1))/log_e2]`
`y=1/log_e2*x^(3/2)*log_esqrt(x+1)`
`y=1/ln2*x^(3/2)*lnsqrt(x+1)`
`(dy)/(dx)=1/ln2[x^(3/2)d/(dx)(lnsqrt(x+1))+lnsqrt(x+1)d/(dx)(x^(3/2))]`
`=1/ln2[x^(3/2)*1/(sqrt(x+1))*1/(2sqrt(x+1))+lnsqrt(x+1)*3/2*x^(1/2)]`
`=1/ln2[(x*sqrtx)/(2(x+1))+3/2*sqrtx*lnsqrt(x+1)]`
`=(sqrtx)/(2ln2)[x/(x+1)+3*lnsqrt(x+1)]`