Continuity and Differentiability

f(x) is continuous at x = a if lim_x→a f(x) = f(a)

d/dx (c) = 0
d/dx (x) = 1
d/dx (xⁿ) = nx^n-1
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
d/dx (cosec x) = -cosec x cot x
d/dx (sec x) = sec x tan x
d/dx (cot x) = -cosec² x
d/dx (e^x) = e^x
d/dx (ln x) = 1/x

d/dx (uv) = u(dv/dx) + v(du/dx)

d/dx (u/v) = [v(du/dx) - u(dv/dx)] / v²

d/dx (f(g(x))) = f'(g(x)) · g'(x)
dy/dx = (dy/du) · (du/dx)

d/dx (sin⁻¹ x) = 1/√(1 - x²)
d/dx (cos⁻¹ x) = -1/√(1 - x²)
d/dx (tan⁻¹ x) = 1/(1 + x²)
d/dx (cosec⁻¹ x) = -1/(|x|√(x² - 1))
d/dx (sec⁻¹ x) = 1/(|x|√(x² - 1))
d/dx (cot⁻¹ x) = -1/(1 + x²)

if y = f(x) and f(x) > 0, then
ln y = ln f(x)
On differentiating both sides,
(1/y) (dy/dx) = (1/f(x)) (df/dx)
dy/dx = y (df/dx) / f(x)