The derivative is a function that is geometrically defined as the slope of the line tangent to the curve at any point. If f is differentiable and continuous at [a,b], then
f'(x) = lim h→0 f(x+h)−f(x)/h
lim h→0 f(x+h)−f(x)h.This change in h is infinitely very small. We denote it by Δx. Then change in the original function f(x) is also small, denoted by Δy. The derivative so obtained by applying the limits is also defined as the instantaneous rate of change of a function with respect to a variable.
f'(x) = Δy/Δx = lim x→0 f(x+δx)−f(x) /δxΔ yΔx =limδx→0f(x+δx)−f(x)δx.
If the derivative > 0, then the curve is increasing, and if the derivative < 0, then the curve is decreasing. The derivative at any stationary point = 0 which implies that the function is neither increasing nor decreasing.