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Question
If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`
Solution
x = `sqrt(1 + u^2)`
Differentiating w.r.t.. 'u',
`(dx)/(du) = (2u)/(2sqrt(1 + u^2)`
= `u/sqrt(1 + u^2)` ......(1)
Now, y = `log(1 + u^2)`
Differentiating w.r.t, u,
`(dy)/(du) = 1/(1 + u^2) * d/(du) (1 + u^2)`
⇒ `(dy)/(du) = (2u)/(1 + u^2)` ......(2)
We have,
`(dy)/(dx) = ((dy)/(du))/((dx)/(du))`
= `(((2u)/(1 + u^2)))/(u/sqrt(1 + u^2))`
= `(2u)/(1 + u^2) xx sqrt(1 + u^2)/u`
= `2/sqrt(1 + u^2)`
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