Implicit Function Derivative

Suppose that there is the dependence between argument х and function у given by the equation, unsolvable with regard to у. Such dependence defines у as the implicit function of х:

F(x;y)=0. (*)

To obtain derivative у in variable х it is required to differentiate equation (*) in х and у, considering у as the composite function of х, i.e., multiplying by y_х. In the obtained equation, we find the similar terms containing y_х. And, solving it as the equation, we obtain the derivative y_х.

Differentiation of the Function Given Parametrically

Suppose that there is function y(х) given parametrically:

Suppose that functions (t) and (t) are differentiable in parameter t and _t0, there is also the inverse function t=t^-1(x). Then the derivative of function can be obtained by the formula: .

Method of Logarithmic Differentiation

The conception of this method is the following: beforehand, we obtain the logarithm of the assigned function, and only then differentiate the result.

In applications more often we encounter with two cases with the logarithmic differentiation.

1. Find the derivative of the product of several functions

y=u₁(х.)· u₂^.(х)· u₃^.(х)· …·u_n(х).

Let us obtain the logarithm of both parts, using the property of the logarithm of product

ln y = ln u₁^.+ ln u₂ + ln u₃ +…+ ln u_n.

Differentiate both parts, the left one - as the implicit function

.

Multiplying both parts by у and substituting it by the function itself, we obtain

.

2. Consider the function

,

here the base u(x) and v(x) the power are functions; thus, the function is called a power-exponential function.

Obtain the derivative of this function. By the analogy with the first case, obtain the logarithm and use the property of the logarithm of power

ln y = v(x)^. ln u(x).

Differentiate both parts

,

multiplying both parts by у and substituting by the power-exponential function; thus, we obtain the derivative

.

2. Differential of a function

Concept of differential

Recall that the derivative of y = f(x) at x is: .

dy = f '(a) dx.

Definitions 2. i.    The differential of x is a change in x, denoted dx. ii.   The differential of the function y = f(x) at x = a is the function defined by dy = f '(a) dx,       where dx is the independent variable and dy the dependent variable. iii.  The differential of the function y = f(x) is the operation which associates to each point x in      dom ( f ) the differential dy = f '(x) dx of y = f(x) at that point.

Use differential to approximate calculations

For x near , is a good approximation for .

or

The error becomes arbitrarily small, compared with , as .

Example. Find the approximate value of the function at x = 1.02.

Solution. Use , х=1. Then . . , we obtain . Therefore, .