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Question
If y = `log((x + sqrt(x^2 + a^2))/(sqrt(x^2 + a^2) - x))`, find `dy/dx`.
Sum
Solution
y = `log((x + sqrt(x^2 + a^2))/(sqrt(x^2 + a^2) - x))`
∴ y = `log(x + sqrt(x^2 + a^2)) - log(sqrt(x^2 + a^2) - x)`
∴ `dy/dx = 1/(x + sqrt(x^2 + a^2))[1 + x/sqrt(x^2 + a^2)] - 1/(sqrt(x^2 + a^2) - x)[x/sqrt(x^2 + a^2) - 1]`
= `(sqrt(x^2 + a^2) + x)/(sqrt(x^2 + a^2)(x + sqrt(x^2 + a^2))) - (x - sqrt(x^2 + a^2))/(sqrt(x^2 + a^2)(sqrt(x^2 + a^2) - x)`
= `1/sqrt(x^2 + a^2) + (sqrt(x^2 + a^2) - x)/(sqrt(x^2 + a^2)(sqrt(x^2 + a^2) - x)`
= `1/sqrt(x^2 + a^2) + 1/sqrt(x^2 + a^2)`
= `2/sqrt(x^2 + a^2)`
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