Basic Derivatives

d/dx f(x) = f'(x)  notation
                   f(x) is any function of x, any equation
                   f'(x) is derivative of f(x) with respect to x

d^n/dx^n f(x) = d^n-1/dx^n-1 f'(x)  reduce nth order derivative

a may be any equation without an x
b may be any equation without an x

d/dx a = 0

d/dx x = 1

d/dx a*x = a   

d/dx a*f(x) = a*f'(x)

d/dx a*f(x)+b*g(x) = a*f'(x) + b*g'(x)

d/dx a*f(x)-b*g(x) = a*f'(x) - b*g'(x)

d/dx f(x)*g(x) = f(x)*g'(x) + f'(x)*g(x)

d/dx f(x)/g(x) = (f'(x)*g(x) - f(x)*g'(x)) / g(x)^2 

d/dx y(z(x)) = d/dz y(z) * d/dx z(x)

d/dx x^n = n*x^n-1

d/dx sin(x) = cos(x)

d/dx cos(x) = -sin(x)

d/dx tan(x) = sec(x)^2  = 1 / cos(x)^2

d/dx cot(x) = -csc(x)^2 = -1 / sin(x)^2

d/dx asin(x) = 1 / sqrt(1-x^2)

d/dx acos(x) = -1 / sqrt(1-x^2)

d/dx atan(x) = 1 / (1+x^2)

d/dx acot(x) = -1 / (1+x^2)

d/dx sinh(x) = cosh(x)

d/dx cosh(x) = sinh(x)

d/dx tanh(x) = 1 / cosh(x)^2

d/dx coth(x) = -1 / sinh(x)^2

d/dx asinh(x) = 1 / sqrt(1+x^2)

d/dx acosh(x) = 1 / sqrt(x-1)*sqrt(x+1)

d/dx atanh(x) = 1 / (1-x^2)

d/dx acoth(x) = 1 / (1-x^2)

d/dx exp(x) = exp(x)

d/dx exp(z(x)) = exp(z(x))*z'(x)

d/dx a^x = ln(a)*a^x

d/dx a^(x^2) = 2*a^(x^2) *x*ln(a)

d/dx ln(x) = 1 / x

d/dx sqrt(x) = 1 / 2*sqrt(x)

d/dx x^(1/2) = 1/2 * x^(-1/2)

d/dx x^(-1) = -1 * x^(-2)