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Question
Differential coefficient of sec(tan−1 x) is ______.
Options
`x/(1 + x^2)`
`x sqrt(1 + x^2)`
`1/sqrt(1 + x^2)`
`x/sqrt(1 + x^2)`
Solution
`x/sqrt(1 + x^2)`
Explanation:
y = sec(tan−1 x)
`dy/dx = d/dx sec(tan^-1 x)`
`dy/dx = sec(tan^-1 x). tan(tan^-1 x) × d/dx (tan^-1 x)`
`dy/dx = sec(tan^-1 x). tan(tan^-1 x) × 1/sqrt(1 + x^2)`
`dy/dx = sec(tan^-1 x). x × 1/sqrt(1 + x^2) ...[tan(tan^-1 x) = x]`
`dy/dx = y × x × 1/sqrt(1 + x^2)`
`dy/dx = y(x/sqrt(1 + x^2))`
`dy/dx = (x/sqrt(1 + x^2))y`
This is the equation of differential equation which have coefficient `x/sqrt(1 + x^2)`.
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