{"generator":"equations.calculus","version":1,"value":{"problems":[{"prompt":"d\/dx [ x*ln(x) ]","prompt_latex":"\\frac{d}{dx}\\left[\\, x \\cdot \\ln\\left(x\\right) \\,\\right]","answer":"ln(x) + 1","answer_latex":"\\ln\\left(x\\right) + 1","rule":"product"},{"prompt":"d\/dx [ x^3\/(x + 6) ]","prompt_latex":"\\frac{d}{dx}\\left[\\, \\frac{x^{3}}{x + 6} \\,\\right]","answer":"(3x^2(x + 6) - x^3)\/(x + 6)^2","answer_latex":"\\frac{3x^{2}\\left(x + 6\\right) - x^{3}}{\\left(x + 6\\right)^{2}}","rule":"quotient"},{"prompt":"d\/dx [ ln(2x + 7) ]","prompt_latex":"\\frac{d}{dx}\\left[\\, \\ln\\left(2x + 7\\right) \\,\\right]","answer":"2\/(2x + 7)","answer_latex":"\\frac{2}{2x + 7}","rule":"chain"},{"prompt":"d\/dx [ sqrt(4x + 1) ]","prompt_latex":"\\frac{d}{dx}\\left[\\, \\sqrt{4x + 1} \\,\\right]","answer":"2\/sqrt(4x + 1)","answer_latex":"\\frac{2}{\\sqrt{4x + 1}}","rule":"chain"},{"prompt":"d\/dx [ sin(5x + 6) ]","prompt_latex":"\\frac{d}{dx}\\left[\\, \\sin\\left(5x + 6\\right) \\,\\right]","answer":"5cos(5x + 6)","answer_latex":"5\\cos\\left(5x + 6\\right)","rule":"chain"},{"prompt":"d\/dx [ 6sin(x) + 4x ]","prompt_latex":"\\frac{d}{dx}\\left[\\, 6\\sin\\left(x\\right) + 4x \\,\\right]","answer":"6cos(x) + 4","answer_latex":"6\\cos\\left(x\\right) + 4","rule":"trig"}],"count":6},"display":"1. d\/dx [ x*ln(x) ]   \u2192   ln(x) + 1\n2. d\/dx [ x^3\/(x + 6) ]   \u2192   (3x^2(x + 6) - x^3)\/(x + 6)^2\n3. d\/dx [ ln(2x + 7) ]   \u2192   2\/(2x + 7)\n4. d\/dx [ sqrt(4x + 1) ]   \u2192   2\/sqrt(4x + 1)\n5. d\/dx [ sin(5x + 6) ]   \u2192   5cos(5x + 6)\n6. d\/dx [ 6sin(x) + 4x ]   \u2192   6cos(x) + 4","meta":{"latex":[["\\frac{d}{dx}\\left[\\, x \\cdot \\ln\\left(x\\right) \\,\\right]","\\ln\\left(x\\right) + 1"],["\\frac{d}{dx}\\left[\\, \\frac{x^{3}}{x + 6} \\,\\right]","\\frac{3x^{2}\\left(x + 6\\right) - x^{3}}{\\left(x + 6\\right)^{2}}"],["\\frac{d}{dx}\\left[\\, \\ln\\left(2x + 7\\right) \\,\\right]","\\frac{2}{2x + 7}"],["\\frac{d}{dx}\\left[\\, \\sqrt{4x + 1} \\,\\right]","\\frac{2}{\\sqrt{4x + 1}}"],["\\frac{d}{dx}\\left[\\, \\sin\\left(5x + 6\\right) \\,\\right]","5\\cos\\left(5x + 6\\right)"],["\\frac{d}{dx}\\left[\\, 6\\sin\\left(x\\right) + 4x \\,\\right]","6\\cos\\left(x\\right) + 4"]],"problems":6,"stream_bytes_used":28,"rejections":5,"entropy_in_bytes":16,"duration_us":1481},"receipt":{"source":"bitcoin","source_label":"Bitcoin Proof-of-Work","class":"B","class_label":"Public beacon","caveat":"Published openly \u2014 anyone can look this value up after the fact.","narrative":"Bitcoin block 966,862 \u2014 19 leading zeros of proof-of-work, found about ten minutes ago.","proof_url":"https:\/\/mempool.space\/block\/0000000000000000000234f6957a807151bafa2cfc8917a02c4d3f9e81cb078e","reference":"966862","observed_at":"2026-09-13T19:21:37+00:00","degraded":false,"mixed_with_csprng":true,"latency_ms":0,"cached":true},"seed":{"token":"KKMXMKG4NQC201ADN172XCJ9PW","permalink":"https:\/\/randomly.cluxnei.dev\/r\/KKMXMKG4NQC201ADN172XCJ9PW?g=equations.calculus&v=1&p=eyJjb3VudCI6NiwibW9kZSI6ImRlcml2YXRpdmUiLCJydWxlIjoibWl4ZWQifQ&s=bitcoin&r=966862"},"uses_entropy":true,"reproducible":true}