Math

Derivative

100

d/dx sin(x²)

2xcos(x²)

200

d/dx arctan(1/x)

-1/(1+x²)

300

d/dx x^sin(x)

(ln(x)cos(x)+sin(x)/x)x^sin(x)

400

d/dx arctan(x)^tan(x)

arctan(x)^tan(x)(tan(x)/(arctan(x)(x²+1))+sec²(x)ln(arctan(x)))

500

d²/dx² x^x

x^x(1/x + (ln(x)+1)²)

Integral

100

Int from 0 to 3 x² dx

9

Sum

200

from 1 to infinity 1/n

infinity

Product

100

from 1 to k n

k!

400

from 0 to infinity F(n+1)/phiF(n) where F(n) is the nth Fibonacci number

(phi+1)/(phi+2) or phi/root(5)

Miscellaneous

400

An a convergent approximation for the lambert w function

^∞(e^-x)x or ln(x/ln(x/ln(x/ln(x...))))

500

real value of x that makes (i-x²)/ln(x) also real

-e^-W(2/pi)/2