The method of differentiation is used to locate function, derived from the original. The derivative of a function is the ratio of the limit of the increment function to increment of argument. This is the most common representation of the derivative, which is denoted by an apostrophe mark "'". Perhaps the repeated differentiation of a function, with the formation of the first derivative f’(x), the second f’(x), etc. denote higher-order Derivatives f^(n)(x).
To differentiate the function, you can use the Leibniz formula:(f*g)^(n) = Σ C(n)^k*f^(n-k)*g^k where C(n)^k– accepted binomial coefficients. The simplest case of the first derivative easier to consider a concrete example: f(x) = x^3.
So, by definition:f’(x) = lim ((f(x) – f(x_0))/(x – x_0)) = lim ((x^3 – x_0^3)/(x – x_0)) = lim ((x – x_0)*(x^2 +x* x_0 + x_0^2)/(x – x_0)) = lim (x^2 + x*x_0 + x_0^2) x seeking to the value of x_0.
Get rid of the sign of the limit, substituting in the obtained expression the value of x equal to x_0. We get:f’(x) = x_0^2 + x_0*x_0 + x_0^2 = 3*x_0^2.
Consider differentiation of complex functions. These functions are compositions or compositions of functions, i.e. the result of one function is the argument to the other:f = f(g(x)).
The derivative of such a function has the form:f’(g(x)) = f’(g(x))*g’(x), i.e., equal to the product of the older of the function on argument the younger younger on the derivative of the function.
To differentiate a composition of three or more functions, apply the same rule in the following way:f’(g(h(x))) = f’(g(h(x)))*(g(h(x)))’ = f’(g(h(x)))*g’(h(x))*h’(x).
Knowledge of the derivatives of some simple functions is a good help in solving problems in differential calculus:- derivative of constant is equal to 0;- the derivative of a simple function of the argument in the first degree x’ = 1;- derivative of sum of functions is equal to the sum of their derivatives: (f(x) + g(x))’ = f’(x) + g’(x);- similarly, the derivative of the product is the product of the derivatives;- the private derivative of two functions: (f(x)/g(x))’ = (f’(x)*g(x) – f(x)*g’(x))/g^2(x);- (C*f(x))’ = C*f’(x), where C is a constant in the differentiation degree of a single term is submitted in the form of a multiplier, and the degree is lowered by 1: (x^a)’ = a*x^(a-1);- the trigonometric functions sinx and cosx in differential calculus are respectively odd and even character - (sinx)’ = cosx and (cosx)’ = - sinx;- (tg x)’ = 1/cos^2 x;- (ctg x)’ = - 1/sin^2 x.