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Derivatives of Algebraic Functions
Introduction to the concept of derivatives of algebraic functions begins at high school level. Where the derivative of algebraic functions is still related to the material on functions that has been studied at junior high school level. Even though the knowledge of deriving algebraic functions has been introduced at high school level, the actual application of deriving algebraic functions is close to our daily lives.
An example is when watching a television broadcast regarding the presidential election, there is a Quick Count to temporarily estimate which candidate will be elected. Well, this is one application of algebraic function derivatives in everyday life. To find out more details about derivatives of algebraic functions, you can read this discussion until the end.
Understanding Algebraic Function Derivatives
If understood carefully, the use of the concept of deriving algebraic functions is closely related to everyday human life. Like when we watch television and use an algebra calculator, this is one of the results of applying algebraic function derivatives in life. Of course, it is very confusing what is meant by the derivative of an algebraic function.
A derivative of a function can be called a differential, which is another function from a function that appeared or was present before. For example, the function f which makes f’ the power of its value does not use rules and the results of the function will not change according to the variables entered and in general a quantity can change due to other things. big change.
Usually a function derivative is defined as a measurement where the results of the function change according to the variables entered. Simply put, a quantity that changes follows changes in other quantities. Meanwhile, the process used to find derivatives is called differentiation.
The derivative is a measurement of how a function changes due to a change in the value entered, indicating that a quantity changes due to a change in another quantity. Functions from derivatives are algebraic forms that are often found to calculate tangents to curves or functions and velocities.
Lines on a graph have a certain gradient which is based on the location of the abscissa or x coordinate and the ordinate or what is called the y coordinate. And where each point is, if the two points are moved closer to each other then the distance between the points also approaches zero then the slope of the graph through the line will change.
At first this line will intersect the curve at two points, then it will turn into a tangent line which seems to touch the curve at only one point. This is because the interpoint is approaching zero, so the quantity can be known through the concept of algebraic function limits.
Algebraic Function Derivative Formula
If you already know the meaning, here is the algebraic function derivative formula that is applied:
f(x) = b f(x) = 0
A constant will have a value of zero if it is reduced, for example f(x) = 15 f(x) = 0.
f(x) = bx f(x) = b
If variable x is reduced relative to x, it will produce 1. Example:
f(x) = x f(x) = 1
f(x) = 2x f(x) = 2
f(x) = 5x 3 f(x) = 5
f(x) = axn f(x) = naxn-1
To better understand, below is an example of an algebraic function derivative question and its discussion.
Examples of Algebraic Function Derivative Questions
- The first derivative of the function f(x) = 3×4 + 2×2 5x is …..?
Discussion:
f(x) = axn f\'(x) = anxn-1
f(x) = 3×4 + 2×2 5x
f\'(x) = 4.3×4-1 + 2.2×2-1 1.5×1-1
f\'(x) = 12×3 + 4×1 5×0
f\'(x) = 12×3 + 4x 5
1. The first derivative of the function f(x) = 2×3 + 7x is …..?
Discussion:
f(x) = axn f\'(x) = anxn-1
f(x) = 2×3 + 7x
f\'(x) = 3.2×3-1 + 1.7×1-1
f\'(x) = 6×2 + 7×0
f\'(x) = 6×2 + 7
2. The derivative of f(x) = x3 is …..?
Discussion:
f(x) = xn f\'(x) = nxn-1
f(x) = x3
f\'(x) = 3×3-1
f\'(x) = 3×2
3. Find the derivative f\'(x) of the function f(x) = 3×2 + 7x ?
Discussion:
f(x) = U + V f\'(x) = U\’ + V\’
f(x) = 3×2 + 7x
From this function we get:
U = 3×2
U\’ = 32×2 1
U\’ = 6x
V= 7x
V\’ = 71×1 1
V\’ = 7
f\'(x) = U\’ + V\’
f\'(x) = 6x + 7
Get to know the application of derivatives of algebraic functions
A derivative or differential function is another function from a previous function, for example the function f'(x) from the original f(x). Differential results can change according to the variables entered or other quantities. Derivatives of algebraic functions can also be applied to various mathematical materials, of course with different concepts too. There are four types of applications of algebraic function derivatives, namely:
Determines the gradient of the tangent line
Algebraic function derivatives can be used to determine the slope or gradient of the tangent line (m) of a curve. On a curve with the formula y = f(x), the gradient can be formulated as: m = y’ = f'(x).
Determine the interval of increasing function and decreasing function
Algebraic function derivatives can determine function intervals with certain conditions. In an increasing function, the interval condition must be f'(x) > 0. Meanwhile, the interval condition for a decreasing function is f'(x) < 0.
Determine stationary values and their types
If the function y = f(x) is continuous and differentiable at x = a and f'(x) = 0, then the function has a stationary value at x = a. The type of stationary value of the function y = f(x) is determined by the second derivative, which can be a minimum return value, a maximum return value, or an inflection value.
Solving limit problems of indefinite form
Derivatives of algebraic functions can also be used to solve limits in the form of indeterminate or 0/0. In this problem, you can use the derivatives of the functions f(x) and g(x) along with their second derivatives. If the first derivative produces a certain shape, then that shape is the solution.
However, if the first derivative still produces an uncertain form, then each f(x) and g(x) are lowered again until the result is a certain form.
So, that’s the definition, formula, example questions, and application of algebraic function derivatives. Hopefully this discussion will be useful.
