Derivative

Definition

The derivative of the function f at a is the limit of the difference quotient as \Delta approaches zero. If the limit exists, then f is differentiable at a.

f'(a) = \lim_{h\to 0} \dfrac{f(a+\Delta)-f(a)}{\Delta}

In calculus, the differential represents the principal part of the change in a function y=f(x) with respect to changes in the independent variable. The differential dy is defined by:

dy = f'(x)dx

where f'(x) is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation

dy = \frac{dy}{dx} dx

holds, where the derivative is represented in the Leibniz notation \frac{dy}{dx}, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes

df(x) = f'(x)dx

The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. In physical applications, the variables dx and dy are often constrained to be very small (“infinitesimal”).

Notations

  • Lagrange’s notation: f'(x)
  • Leibnitz’s notation: \frac{df}{dx}(a)

Derivative rules

The Constant Rule

The derivative of a constant function is 0. That is, if c is a real number, then

\dfrac{d}{dx}(c) = 0

The Sum and Difference Rules

The sum(or difference) of two differentiable functions is differentiable and is the sum(or difference) of their derivatives.

\dfrac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)
\dfrac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)

The Constant Multiple Rule

If f is a differentiable function and c is a real number, then cf is also differentiable.

\dfrac{d}{dx}(cf(x)) = cf'(x)

The Power Rule

If n is a rational number, then the function f(x) = x^n is differentiable.

\dfrac{d}{dx}(x^n) = n\cdot x^{n-1}

The Product Rule

The product of two differentiable functions, f and g, is itself differentiable. Moreover, the derivative of f\cdot g is the first function times the derivative of the second, plus the second function times the derivative of the first.

\dfrac{d}{dx}(f(x)g(x)) = f(x)g'(x) + g(x)f'(x)

The Quotient Rule

The quotient \frac{f}{g}, of two differentiable functions, f and g, is itself differentiable at all values of x for which g(x)\neq 0. Moreover, the derivative of \frac{f}{g} is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the square of the denominator.

\dfrac{d}{dx}( \dfrac{f(x)}{g(x)} ) = \dfrac{g(x)f'(x) - f(x)g'(x)}{ g(x)^2 }, g(x) \neq 0

The Chain Rule

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and

\dfrac{d}{dx}(f(g(x))) = f'(g(x))g'(x)

The General Power Rule

If y = u(x)^n, where u is a differentiable function of x and n is a rational number, then

\dfrac{d}{dx}(u(x)^n) = n\cdot u^{n-1}\cdot u'

Derivatives of common functions

  • (x^\alpha)' = \alpha \cdot x^{\alpha - 1} \ (\alpha = const)
  • (x^x)' = x^x (\ln x + 1)
  • (\sin x)' = \cos x
  • (\cos x)' = - \sin x
  • (\tg x)' \frac{1}{\cos^2 x}
  • (\ctg x)' = - \frac{1}{\sin^2 x}
  • (\arcsin x)' = \frac{1}{\sqrt{1 - x^2}}
  • (\arccos x)' = - \frac{1}{\sqrt{1 - x^2}}
  • (\arctg x)' = \frac{1}{1 + x^2}
  • (\arcctg x)' = - \frac{1}{1 + x^2}
  • (a^x)' = a^x \ln a
  • (\log_a x)' = \frac{1}{x \ln a}
  • (\sh x)' = \ch x
  • (\ch x)' = \sh x
  • (\th x)' = \frac{1}{ch^2 x}
  • (\cth x)' = - \frac{1}{sh^2 x}
  • (e^x)^{\prime} = e^x
  • \ln x^{\prime} = \frac{1}{x}
  • (\sqrt{x})^\prime = \frac{1}{2 \sqrt{x}}