Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

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

Answer 1

`(dy)/(dx)=\ {(1)/(2)}*((1)/((x^2 - 1 +x))}*((x^2 +1)/(x))})`

Explanation:

Given function:

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

now,

ln(AB) = ln(A) + ln(B)

thus,

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

also,

ln(Aⁿ) = n × ln(A)

therefore,

y = ln(3) +
`{(1)/(2)}\ln(x - (1)/(x)+1})`

differentiating with respect to 'x'

`(dy)/(dx)=0\ +\ {(1)/(2)}*((1)/((x - (1)/(x)+1)}*(1 - (-1*(1)/(x^2))+0})`

or

`(dy)/(dx)=0\ +\ {(1)/(2)}*((1)/(((x^2 - 1 +x)/(x)))}*(1 +(1)/(x^2))})`

or

`(dy)/(dx)=\ {(1)/(2)}*((x)/((x^2 - 1 +x))}*((x^2 +1)/(x^2))})`

or

`(dy)/(dx)=\ {(1)/(2)}*((1)/((x^2 - 1 +x))}*((x^2 +1)/(x))})`