d\/dx((x sin^(-1)(x))\/sqrt(1-x^2)+log(sqrt(1-x^2))) = (sin^(-1)(x))\/(1-x^2)^(3\/2)
Possible intermediate steps:
Possible derivation:\nd\/dx((x sin^(-1)(x))\/sqrt(1-x^2)+log(sqrt(1-x^2)))\nDifferentiate the sum term by term:\n = d\/dx((x sin^(-1)(x))\/sqrt(1-x^2))+d\/dx(log(sqrt(1-x^2)))\nUse the product rule, d\/dx(u v) = v ( du)\/( dx)+u ( dv)\/( dx), where u = x and v = (sin^(-1)(x))\/sqrt(1-x^2):\n = d\/dx(log(sqrt(1-x^2)))+(sin^(-1)(x) d\/dx(x))\/sqrt(1-x^2)+x d\/dx((sin^(-1)(x))\/sqrt(1-x^2))\nSimplify the expression:\n = (sin^(-1)(x) (d\/dx(x)))\/sqrt(1-x^2)+x (d\/dx((sin^(-1)(x))\/sqrt(1-x^2)))+d\/dx(log(sqrt(1-x^2)))\nThe derivative of x is 1:\n = x (d\/dx((sin^(-1)(x))\/sqrt(1-x^2)))+d\/dx(log(sqrt(1-x^2)))+(1 sin^(-1)(x))\/sqrt(1-x^2)\nUse the product rule, d\/dx(u v) = v ( du)\/( dx)+u ( dv)\/( dx), where u = 1\/sqrt(1-x^2) and v = sin^(-1)(x):\n = (sin^(-1)(x))\/sqrt(1-x^2)+d\/dx(log(sqrt(1-x^2)))+sin^(-1)(x) d\/dx(1\/sqrt(1-x^2))+(d\/dx(sin^(-1)(x)))\/sqrt(1-x^2) x\nUsing the chain rule, d\/dx(1\/sqrt(1-x^2)) = d\/( du)1\/sqrt(u) ( du)\/( dx), where u = 1-x^2 and ( d)\/( du)(1\/sqrt(u)) = -1\/(2 u^(3\/2)):\n = (sin^(-1)(x))\/sqrt(1-x^2)+d\/dx(log(sqrt(1-x^2)))+x ((d\/dx(sin^(-1)(x)))\/sqrt(1-x^2)+(-d\/dx(1-x^2))\/(2 (1-x^2)^(3\/2)) sin^(-1)(x))\nDifferentiate the sum term by term and factor out constants:\n = (sin^(-1)(x))\/sqrt(1-x^2)+d\/dx(log(sqrt(1-x^2)))+x ((d\/dx(sin^(-1)(x)))\/sqrt(1-x^2)-(d\/dx(1)-d\/dx(x^2) sin^(-1)(x))\/(2 (1-x^2)^(3\/2)))\nThe derivative of 1 is zero:\n = (sin^(-1)(x))\/sqrt(1-x^2)+d\/dx(log(sqrt(1-x^2)))+x ((d\/dx(sin^(-1)(x)))\/sqrt(1-x^2)-(sin^(-1)(x) (-(d\/dx(x^2))+0))\/(2 (1-x^2)^(3\/2)))\nSimplify the expression:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x ((sin^(-1)(x) (d\/dx(x^2)))\/(2 (1-x^2)^(3\/2))+(d\/dx(sin^(-1)(x)))\/sqrt(1-x^2))+d\/dx(log(sqrt(1-x^2)))\nUse the power rule, d\/dx(x^n) = n x^(n-1), where n = 2: d\/dx(x^2) = 2 x:\n = (sin^(-1)(x))\/sqrt(1-x^2)+d\/dx(log(sqrt(1-x^2)))+x ((d\/dx(sin^(-1)(x)))\/sqrt(1-x^2)+(2 x sin^(-1)(x))\/(2 (1-x^2)^(3\/2)))\nSimplify the expression:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x ((x sin^(-1)(x))\/(1-x^2)^(3\/2)+(d\/dx(sin^(-1)(x)))\/sqrt(1-x^2))+d\/dx(log(sqrt(1-x^2)))\n
sin^(-1)(x) is 1\/sqrt(1-x^2):\n = (sin^(-1)(x))\/sqrt(1-x^2)+d\/dx(log(sqrt(1-x^2)))+x ((x sin^(-1)(x))\/(1-x^2)^(3\/2)+1\/sqrt(1-x^2)\/sqrt(1-x^2))\n
\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))+d\/dx(log(sqrt(1-x^2)))\nSimplify log(sqrt(1-x^2))
log(a^b) = b log(a):\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))+d\/dx(1\/2 log(1-x^2))\nFactor out constants:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))+(d\/dx(log(1-x^2)))\/2\nUsing the chain rule, d\/dx(log(1-x^2)) = ( dlog(u))\/( du) ( du)\/( dx), where u = 1-x^2 and ( d)\/( du)(log(u)) = 1\/u:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))+1\/2(d\/dx(1-x^2))\/(1-x^2)\nDifferentiate the sum term by term and factor out constants:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))+d\/dx(1)-d\/dx(x^2)\/(2 (1-x^2))\nThe derivative of 1 is zero:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))+(-(d\/dx(x^2))+0)\/(2 (1-x^2))\nSimplify the expression:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))-(d\/dx(x^2))\/(2 (1-x^2))\nUse the power rule, d\/dx(x^n) = n x^(n-1), where n = 2: d\/dx(x^2) = 2 x:\n = (sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))-2 x\/(2 (1-x^2))\nSimplify the expression:\nAnswer: | \n | = -x\/(1-x^2)+(sin^(-1)(x))\/sqrt(1-x^2)+x (1\/(1-x^2)+(x sin^(-1)(x))\/(1-x^2)^(3\/2))