Question

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

y =In x/x^3

Answer 1

Answer:
`(1-3ln x)/(x^(4))`

Explanation:

According to Quotient rule in derivatives :

If f(x) and g(x) are differentiable functions , then

`(d)/(dx)((f(x))/(g(x)))=(f'(x)g(x)-f(x)g'(x))/(g(x)^2)`

The given function :
`y=(\ln x)/(x^3)`

Differentiate both sides , we get

`(dy)/(dx)=(d)/(dx)((\ln x)/(x^3))`

By Quotient rule , this is equal to

`=((\ln x)'(x^3)-(x^3)'(\lnx))/((x^3)^2)`

`=(((1)/(x))(x^3)-\ln x(3x^2))/(x^6)`

[∵
`(d)/(dx)(\ln x)=(1)/(x)\ \ \&\ (d)/(dx)(x^n)=n(x)^(n-1)`]

`=(x^2-3x^2\ln x)/(x^6) \\\\=(x^2(1-3\ln x))/(x^6)\\\\=((1-3\ln x))/(x^(6-2))\\\\=(1-3\ln x)/(x^(4))`

Hence, the derivative of the function will be :
`(1-3ln x)/(x^(4))`