Derivative from first principles

Letting the point vary gives a new function, the derivative of f: f′(x) = lim_{h→0} [f(x + h) − f(x)]/h. Finding it straight from this limit is called working from first principles.

If y = f(x), the derivative is written f′(x), d/dx f(x), dy/dx or D(f(x)). Its value at x = a is written f′(a) or (dy/dx) at x = a.

Worked: f(x) = 10x gives [10(x + h) − 10x]/h = 10, so f′(x) = 10.

Worked: f(x) = x² gives (2xh + h²)/h = 2x + h → 2x.

Worked: a constant f(x) = a gives (a − a)/h = 0, so f′(x) = 0.

Worked: f(x) = 1/x gives [1/(x + h) − 1/x]/h = −1/[x(x + h)] → −1/x², for x ≠ 0.