Differentiation is the process of finding a derivative in calculus. How a function changes with respect to changes of some input is a broad definition of a derivative. Velocity is an example of a derivative; it is a measure of change of position of a moving object with respect to time. Mathematically the derivative of a function is given by:
f'(x) = limit as h approaches 0 of [f(x+h) -- f(x)] / h. This is true if this limit exists. If f'(a) exists, then f is said to be differentiable as x = a. The process of finding the derivative is called differentiation.
Integrals along with derivatives are the basic tools of calculus. Its formal definition is based on approximating the area under a curve by means of breaking the region into a series of rectangles. Mathematically if given a function f(x) and an interval [a,b] on which the function is continuous the definite integral is defined as:
The integral from a to b of f(x) dx = F(b) -- F(a) where F is any function such that F'(x) = f(x) for all x in [a,b].
The mathematical definition of a partial derivative of a function of two variables is, If z = f(x,y), then the partial derivative of z with respect to x at (x,y) is dz/dx = limit as h approaches 0 of [f(x+h, y) -- f(x,y)] / h if this limit exists. The partial derivative of z with respect to y at (x,y) is dz/dy = limit as h approaches 0 [ f(x, y+h) -- f(x,y)] / h if this limit exists. When the partial derivative of z is taken with respect to x, the y is unchanged in the two terms of the numerator; also the x is unchanged when the partial derivative is with respect to y. To find a partial derivative with respect to x, differentiate with respect to x, with y constant. To find the partial derivative with respect to y regard x as constant and differentiate with respect to y. Because thermodynamic systems are characterized by quantitative variables the use of partial differentiation makes it possible to vary one variable with respect to others.
Differential equations allow for the development of mathematical models. Thermodynamic systems may be modeled mathematically with the use of differential equations. Differential equations relate unknown functions to some of their derivatives. A differential equation involving ordinary derivatives with respect to single independent variables is called an ordinary differential equation. An example of a differential equation is found in electrical engineering and Kirchhoff's law. It leads to the equation,
L (d^2q) / (dt^2) + R (dq/dt) I+ (1/C)q = E(t)
where L is the inductance, R is the resistance C is the capacitance, E(t) is the electromotive force, q(t) is the charge and t is the time.