Question

Use the logarithmic differentiation to find the derivative of `y=(xsqrt(x^2+1))/ (x+1)^(2/3)` can any one help me to solve it??

Answer 1

Given: `y=(xsqrt(x^2+1))/ (x+1)^(2/3)`

Use the natural logarithm on both sides:

`ln(y)=ln((xsqrt(x^2+1))/ (x+1)^(2/3))`

We can use the rules of logarithms to separate the right side into terms:

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

Next, we take advantage of the way that exponents can be turned into coefficients:

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

Every term is easy to differentiate:

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

Multiply both sides by y:

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

We are given an equation for y:

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