**Derivatives of Algebraic Functions High School Mathematics Lesson**

Being one of the sciences studied and often found in high school, derivatives of algebraic functions include material that is closely related to function limits and the slope of a line or what is known as a gradient. Derivatives are actually used to show and demonstrate the existence of a change. It might sound very confusing, therefore, let’s look at the complete discussion below.

**Understanding Algebraic Function Derivatives**

If understood carefully, the application of the concept of deriving algebraic functions is very closely related to everyday human life. Like when we watch television and use an algebra calculator, this is one of the results of applying derivatives of algebraic functions in life. Of course, it is very surprising, then, what is meant by the derivative of an algebraic function.

A derivative of a function, also called a differential, is another function from a function that appears or existed before. For example, the function f which is made into f’ with the strength of the value does not use rules and the results of the function do not change, are adjusted to the variables entered and in general a quantity can change due to other large changes.

In general, function derivatives are defined as measurements where the results of the function change according to the variables that have been entered. Simply put, a quantity that changes will follow changes in other quantities. Meanwhile, the process used or applied to find a derivative is called differentiation.

The derivative is a measurement related to how a function can change due to a change in the entered value, indicating that a quantity changes due to a change in another quantity. Functions of algebraic derivatives are those 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 also called the y coordinate. At each point, 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 the beginning the line will intersect the curve at two points, then turn into a tangent line that appears to touch the curve at only one point. This is because the interpoint is approaching zero, then the quantity can be determined through the concept of algebraic function limits which applies the algebraic formula as follows.

**Algebraic Function Derivative Formula**

Derivative equations containing effective limit functions are used for equations of linear functions and powers of 1. However, this formula is less effective if used for algebraic function equations whose polynomial degree is more than 1 (power more than 1). Therefore, you can use the formulas below.

F(x) = b → f‘(x) = 0

A constant is zero if it is lowered, for example f(x) = 15 → f'(x) = 0.

F(x) = bx → f‘(x) = b

If variable x is reduced with respect to x, it produces 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

The formula above applies to derivatives of power functions. When you derive a function, it means you are looking for a derivative of the power of the function or its power becomes smaller. For example, if the variable x2 is reduced with respect to x, then the degree of the variable can be reduced by 1 to x. If the variable x3 is reduced relative to x, then the degree of the variable can be reduced by 1 to become x2 and so on. Notice the example below.

F(x) = 6×4 + 2×3 → f‘(x) = (4)(6)x3 + (3)(2)x2

= 24×3 + 6×2

**Examples of Algebraic Function Derivative Questions**

Example 1

If f(x) = x² – (1/x) + 1, then f'(x) = . . . .

A.

Discussion:

F(x) = x2 – (1/x) + 1

= x2 – x-1 + 1

F'(x) = 2x –(-1)x-1-1

= 2x + x-2

(Answer: E)

Example 2

Y = (x² + 1)(x³ – 1) then y’ is . . . . .

- 5x³ B. 3x³ + 3x C. 2x⁴ – 2x D. x⁴ + x² – x E. 5x⁴ + 3x² – 2x

Discussion:

Y = (x² + 1)(x³ – 1) = x⁵ + x³ – x² – 1

Y’ = 5x⁴ + 3x² – 2x — Answer: E

Example 3

The first derivative of f(x) = (2 – 6x)³ is f'(x) = . . . . .

- -18(2 – 6x)²
- -½(2 – 6x)²
- 3(2 – 6x)²
- 18(2 – 6x)²
- ½(2 – 6x)²

Discussion:

For example: u(x) = 2 – 6x, then u'(x) = -6

F(x) = (u(x))³

F'(x) = 3(u(x))² . u'(x)

= 3(2 – 6x)² . (-6)

= -18(2 – 6x)² —— Answer: A

So, that’s an explanation of the derivatives of algebraic functions, starting from the definition, formula and example questions. By reading the discussion above, it is hoped that students will be able to digest this material well.