Function can be differentiated at any values of an argument, can have a derivative only on certain intervals or not have a derivative at all. But if function has a derivative in some point - it is always number, but not mathematical expression.
Instruction
1. If function Y of one argument x is set in the form of dependence of Y = by F (x), determine its first derivative Y' = by F'(x) by rules of differentiation. To find a function derivative in a certain point x ₀, previously consider area of permissible values of an argument. If x ₀ belongs to this area, then substitute value x ₀ in expression of F'(x) and define required value Y'.
2. Geometrically the function derivative in a point is defined as a tangent of angle between the positive direction of abscissa axis and a tangent to a function graph in a contact point. Tangent is a straight line, and the straight line equation in a general view registers as y=kx +a. A point of contact x ₀ the general for two schedules - function and a tangent. Therefore, Y (x ₀) = y (x ₀). K coefficient is also value of a derivative in the set Y point' (x ₀).
3. If the studied function is set in a graphic look on the coordinate plane, then for finding of derivative function in the necessary point carry out through this point a tangent to a function graph. The tangent is a limit provision of a secant at the maximum rapprochement of points of intersection of a secant with the schedule of the set function. It is known that the tangent is perpendicular to the radius of curvature of the schedule in a contact point. In the absence of other basic data of knowledge of properties of a tangent will help to draw it with bigger reliability.
4. The tangent piece from a point of contact of the schedule before crossing with abscissa axis forms a hypotenuse of a rectangular triangle. One of legs — ordinate of the set point, another — an axis piece OH from a point of intersection from a tangent to a projection of the studied point to an axis OH. The tangent inclination tangent of angle to an axis OH is defined as the relation of an opposite leg (contact point ordinates) to adjacent. The received number is required value of derivative function in the set point.