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Differentiating functions in SageMath
http://doc.sagemath.org/html/en/tutorial/tour_algebra.html#solving-differential-equations
︡9eafe483-de15-4ba0-8762-0a7dc68a0289︡{"done":true,"md":"\nDifferentiating functions in SageMath\n\nhttp://doc.sagemath.org/html/en/tutorial/tour_algebra.html#solving-differential-equations"}
︠50b159c4-8686-4461-9910-626a497ace0cs︠
x = var('x')
a = function('a')(x)
b = function('b')(x)
ab1 = diff(a * b, x)
show(ab1)
︡52b1c172-b45e-4765-8d1e-25c8051d3b0b︡{"html":"
$\\displaystyle b\\left(x\\right) \\frac{\\partial}{\\partial x}a\\left(x\\right) + a\\left(x\\right) \\frac{\\partial}{\\partial x}b\\left(x\\right)$
"}︡{"done":true}
︠051720f2-9298-454a-8829-83e52539118as︠
show(diff(2 * a + 3 * b^2, x))
︡87e7f165-3166-426d-9738-0a185e8312ff︡{"html":"$\\displaystyle 6 \\, b\\left(x\\right) \\frac{\\partial}{\\partial x}b\\left(x\\right) + 2 \\, \\frac{\\partial}{\\partial x}a\\left(x\\right)$
"}︡{"done":true}
︠8559f4c6-4852-49a8-8b22-195c579676a3s︠
# chain rule.
# be explicit about what x is, i.e. a(x= ... )
t = var('t')
show(diff(a(x = t^2 + b(x = x^2)), x))
︡244c08b7-e841-466e-87e7-8afca3abf521︡{"html":"$\\displaystyle 2 \\, x \\mathrm{D}_{0}\\left(a\\right)\\left(t^{2} + b\\left(x^{2}\\right)\\right) \\mathrm{D}_{0}\\left(b\\right)\\left(x^{2}\\right)$
"}︡{"done":true}
︠9f273c20-921a-4d61-8d8d-8a8734b15868s︠
ex3 = (a - 2*b)^3 / (b(x = 2*a(x=x)))^2
show(ex3)
︡0b180b84-39fc-4ddc-b346-852037af412a︡{"html":"$\\displaystyle \\frac{{\\left(a\\left(x\\right) - 2 \\, b\\left(x\\right)\\right)}^{3}}{b\\left(2 \\, a\\left(x\\right)\\right)^{2}}$
"}︡{"done":true}
︠75336b99-5e32-4776-939d-a6387192cd62s︠
show(diff(ex3, x))
︡98d7643b-7ca9-4cd8-9a94-e653c59023b4︡{"html":"$\\displaystyle -\\frac{4 \\, {\\left(a\\left(x\\right) - 2 \\, b\\left(x\\right)\\right)}^{3} \\frac{\\partial}{\\partial x}a\\left(x\\right) \\mathrm{D}_{0}\\left(b\\right)\\left(2 \\, a\\left(x\\right)\\right)}{b\\left(2 \\, a\\left(x\\right)\\right)^{3}} + \\frac{3 \\, {\\left(a\\left(x\\right) - 2 \\, b\\left(x\\right)\\right)}^{2} {\\left(\\frac{\\partial}{\\partial x}a\\left(x\\right) - 2 \\, \\frac{\\partial}{\\partial x}b\\left(x\\right)\\right)}}{b\\left(2 \\, a\\left(x\\right)\\right)^{2}}$
"}︡{"done":true}
︠0063194f-d93a-4673-8270-5b329fd3295ds︠
︡a529cd93-e1d5-484f-aaa9-66eb418df29b︡{"done":true}