DIFFERENSIAL
Standard Competence
6. Use the concept of Function limit and the derived function in problem solving.
Basic Competence
6.3Use the concept and the order of differential in calculation of the derived function.
Indicators
1. Count the simple derived function by using differential definition
2. Explaining the physic meaning of differential at any point
3. Explaining the geometry meaning of differential at any point
4. Determine the change velocity of function value to independent variable
5. Use the order of differential to determine the derived algebra function and trigonometry function
6. Determine the derived composition by chain rule
7. Determine the tangent line equation at any curve
1st meeting (3 x 45 minutes)
Pretest
1. Determine the function h if we have the composition function where and
2. Determine ,for
A. Algebraic Function Derivative
1. The Definition of Derivative
Average velocity
In this case, s is the function of time (t), for example f (t). In the previous chapter, you have learned that speed in , is declared with the following formula.
The formula is called the change rate of distance toward time.
Now, observe the figure on the side. The gradient of the line passing through the point A and B, for example .
if the point B is approaching then the gradient of line becomes the gradient of curve tangent in point A as point B will be coinciding with point A. therefore, if point B A then
For example the function is given. The derivative of this function toward , written as or can be determined use the following formula.
If the limit exist.
In compliance with the concept of the change rate of distance toward, a definition of derivative of a function is made as follows.
The function is given, the function derivative is the function of which value in point c is if the limit value exist.
Further to determine we perceive as a constant, but random so that we would obtain the function derivative of is as follow.
Aside from the notation above, the derivative of function f(x) can be written by the notation this notation is called the Leibniz notation as it was initially suggested by a German mathematician named Gottfried Wilhelm Leibniz (1646-1716).
Example:
Determine the function derivative of
Solution:
1. Using the derivative formula , determine the following function derivative.
a.
b.
c.
d.
e.
f.
2. Using the derivative formula , determine the following function derivative function in .
a.
b.
c.
d.
e.
f.
3. We have the function
a. Determine the function derivative.
b. Determine the value of x so that
2nd meeting (2x45 minutes)
The formula of Function Derivative
Using the formula of function derivative we can determine the derivative of constant function, the derivative of identity function, the derivative of exponent function , and the formula of function derivative with real constant.
a. The constant function derivative , where c is the real constant.