Process of finding of derivative function is called differentiation. The same function can at some values of an argument have a derivative, and at others — not to have.
1. Before looking for a derivative of function it is necessary to explore the area of values of an argument and to exclude those intervals at which existence of function is impossible. For example, for the f=1/x function the value of an argument x =0 is unacceptable, and for the z=loga x function only positive values of an argument are admissible.
2. Derivatives of simple functions of one argument are on differentiation formulas which can be remembered or if necessary to find in tables of derivative elementary functions. For example, the derivative of a constant is always equal to zero, the derivative of linear function f (x) =kx is equal to k coefficient: f' (x) =k, the f (x) function = x² has derivative f' (x) =2x.
3. At differentiation, rules, the general for any function work: - a constant multiplier it is possible to take out for the sign of a derivative: (k*f(x)) of '=k * (f(x))'; - the derivative of the sum of several functions of the same argument is equal to the sum of derivative these functions: (z(x) + f(x)) '=z' (x) +f' (x); - the derivative of performing two functions is equal to the sum of works of derivative first function on the second function and the first function on a derivative of the second function: (z (x) *f(x)) '=z' (x) *f(x) + z (x) *f' (x); - the derivative of private two functions looks so: (z/f)' = (z '*f-z*f') / f².
4. Before applying these rules at differentiation of difficult function, it makes sense to try to simplify initial expression. For example, if it is necessary to find a fraction derivative with a polynomial in numerator, it is possible to divide numerator into a denominator term by term. Then finding of a derivative private functions is replaced with calculation of the derivative algebraic sum of functions. Of course, everyone composed in the received expression will remain in fraction, and it is necessary to find a derivative private, but expressions will be less bulky, and process of differentiation will significantly become simpler. For calculation of value derivative of function in a concrete point, in the received answer instead of an argument x substitute its numerical value and calculate expression.