Differentiation of functions in calculus is the process of finding a new function that represents the rate at which values of y are changing for given values of x along each point of the original curve. There are several major rules to follow when differentiating functions, among them the power rule, quotient rule and chain rule. The general derivative of a basic polynomial function of the form x to the power of n (x^n) is simply the exponent n brought in front of the x value as a coefficient, and n-1 taking the place of the exponent (nx^(n-1)). Derivatives are throughout calculus and physics; you can apply rules of differentiation to vectors and trigonometric functions.
Write out the full function to be differentiated on a piece of paper. Leave plenty of space between terms in the function and below the equation so you have room to work. It is helpful to write out each step as you perform it.
Examine the function for terms that can be easily differentiated by using the product rule. Each term in a function can be differentiated separately, as long as it is not part of a composition of functions. Independent terms that are of the form ax^n, where "a" is a numeric coefficient and "n" is a power, can be readily differentiated using the power rule and should be differentiated first. The derivative will follow the form of n times the constant a times x, all to the power of one less than the original n.
Use the quotient rule to differentiate any terms where there is a variable in the denominator of the term. For example, a function of the form (x + 1)/(x - 1) must be differentiated using the quotient rule, which follows the general form of ((f' * g) - (f * g'))/(g^2) where f is the numerator, f' is the derivative of the numerator, g is the denominator and g' is the derivative of the denominator. The derivatives f' and g' are found using the power rule.
Apply the chain rule to differentiate functions that are compositions. These are functions where there is a variable both on the inside and outside of a particular set of parentheses, which means that there are in fact multiple subfunctions being differentiated. For example, in the expression sin(x^2), there are two functions: x is being squared and the sine of x is being taken. The chain rule specifies that, for these functions, the derivative of the whole function is (f' * g) + (f * g') where f is the inner function, f' is the derivative of the inner function, g is the outer function, and g' is the derivative of the outer function.
Find any special case derivatives that are present in the function and differentiate them using the particular differentiation rules that apply. For example, the derivative of any constant is simply 0. The derivative of the exponential e^x is simply e^x. The derivative of the natural logarithm of x is 1/x. Every operation in calculus is accompanied by a rule of differentiation.