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Calculus question?
Derive arcsine x = ln(x+sqrt(x^2 +1)). Explain in your derivation why plus sign is used with sqrt instead of minus
1 Answer
- cidyahLv 74 years agoFavorite Answer
arcsin x = ln ( x + sqrt(x^2+1) )
Do you want the derivative of arcsin x - ln(x+sqrt(x^2+1))? If so:
differentiate with respect to x
1/sqrt(1-x^2) - 1/(x+sqrt(x^2+1)) d/dx ( x+ sqrt(x^2+1))
d/dx (x+ sqrt(x^2+1)) = 1 + 2x/(2sqrt(x^2+1)) = 1 + x/sqrt(x^2+1)
1/sqrt(1-x^2)) - (1 + x/sqrt(x^2+1) )
= 1/sqrt(1-x^2) - (1 + x/sqrt(x^2+1))/(x+sqrt(x^2+1))
= 1/sqrt(1-x^2) - 1/(x+sqrt(x^2+1)) - x / (xsqrt(x^2+1)+x^2+1)
1+x^2 is always positive. log of a positive number > 1. That's why plus sign is used with sqrt instead of minus.
