## Calculus

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 f Taylor expansion, f(c+x) = f(c) + (f'(c)/1!).x + (f''(c)/2!).x2 + (f'''(c)/3!).x3 + ... 1 / (1+x)r 1 - r x + r.(r+1)/2! x2 - r.(r+1).(r+2)/3! x3 + ... 1 / (1 - x)r 1 + r x + r.(r+1)/2! x2 + ... log(1+x) x - 1/2 x2 + 1/3 x3 - ... log(1 - x) - x - 1/2 x2 - 1/3 x3 - ... sin x x - x3/3! + x5/5! - x7/7! + ... cos x 1 - x2/2! + x4/4! - x6/6! + ... sinh x 1 + x3/3! + x5/5! + ... cosh x 1 + x2/2! + x4/4! + ...

f(x) derivative, d/dx f(x)
c 0
xn n.xn-1
loge x 1/x
x-n -n.x-n-1
ex ex
xx xx(ln(x) + 1)
sin x cos x
cos x - sin x
tan x 1/cos2x = 1+tan2x
sinh x cosh x
cosh x sinh x
tanh x 1 - tanh2 x = 1/cosh2 x
f(x)+g(x) f' + g',   where f'=d/dx f, & g'=d/dx g
f(x) g(x) f'g + fg'
f(x)/g(x) (f'g - fg')/g2
f(g(x)) f'(g(x)) g'(x),   the chain rule
f -1x 1/(f'(f -1x)),   if f -1 is the inverse function of f,   x=f y,   f -1(f(y))=y,   f(f -1x)=x

f(x) indefinite integral f(x) dx
xn (1/(n+1)).xn+1,   n≠-1
x-1 log x

f(x) indefinite f (x) dx -∞+∞ f(x) dx
1 / (a+x2)k x .{hg([1/2,k], [3/2], -x2/a)} / ak √π.Γ(k - 1/2) / {a(k-1/2).Γk}
x2 / (a+x2)k ? √π.Γ(k - 3/2) / {2.a(k-3/2).Γk}

meaning
hg() hypergeometric()
Γ Γ function; for int n, Γn=(n-1)!