State equation of first principles
dy/dx = limdx->0 y(x+dx) - y(x) / dx
State heaviside step function and its differentiation
d(x) = 0 if |x| > 0 but integrate-inf to inf d(x) dx = 1
When is something differentiable at x
continuous function
limit is finite and defined
Differentiate cosh(ax)
a sinh ax
Differentiate cos-1
-1/sqrt(1-x2)
Differentiate tan-1
1/1+x2
Differentiate sin-1
1/sqrt(1-x2)
Differentiate sec, cosec
sectan
-csccot
Differentiate cot, tan
-csc2, sec2
State Leibnitz’s formula, prove
dn(fg)/dx^n = sum m=0 to n (n choose m) f^(n-m) g^m
Proof relation of Hermite polynomials
Hn+1 = 2xHn - 2nHn-1
Hn = (-1)^n e^x2 d^n (e^-x2)/dx^n
How to find points of inflexion
d2y/dx2=0
6 points to take note of when curve sketching
intercepts, symmetries, x to inf, singularities, stationary points, curvature, inflexion points
Draw y = e^ (-x^2/a^2)
normal distribution, width scales with a
