Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function

y = In (x^3 + 1)^1/3

Answer 1

Answer:
`(x^2)/((x^3+1))`

Explanation

Properties of derivative , we use here :

• 
  `(d)/(dx)(\ln x)=(1)/(x)`
• 
  `(d)/(dx)(x^n)=n(x)^(n-1)`
• 
  `(d)/(dx)(a)=0` , where a is constant.

The given function :
`y=\ln(x^3+1)^{(1)/(3)}`

`y=(1)/(3)\cdot \ln (x^3+1)` [
`\because n\ln x=\ln x^n`]

Now , Differentiate both sides , we get

`(dy)/(dx)=(1)/(3)\cdot (1)/((x^3+1))\cdot (d)/(dx)(x^3+1)`

(By chain rule)

`=(1)/(3)\cdot (1)/((x^3+1))\cdot (3x^2+0)`

`=(1)/(3)\cdot (1)/((x^3+1))(3x^2)`

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

Hence, the derivative of the function will be :
`(x^2)/((x^3+1))`