- The derivative as slope of the tangent line to a curve
- The derivative as the instantaneous velocity of a moving object
- Algebraic function
- Bypass rules
- Derivative of a constant
- Derivative of a power
- Derivative of addition and subtraction
- Derived from a product
- Derivative of a quotient
- Chain rule
- References
The algebraic derivatives consist of the study of the derivative in the case of algebraic functions. The origin of the notion of derivative dates back to Ancient Greece. The development of this notion was motivated by the need to solve two important problems, one in physics and the other in mathematics.
In physics, the derivative solves the problem of determining the instantaneous velocity of a moving object. In mathematics, it allows you to find the tangent line to a curve at a given point.
Although there are really many more problems that are solved by using the derivative, as well as its generalizations, results that came after the introduction of its concept.
The pioneers of differential calculus are Newton and Leibniz. Before giving the formal definition, we are going to develop the idea behind it, from a mathematical and physical point of view.
The derivative as slope of the tangent line to a curve
Suppose that the graph of a function y = f (x) is a continuous graph (without peaks or vertices or gaps), and let A = (a, f (a)) be a fixed point on it. We want to find the equation of the line tangent to the graph of the function f at point A.
Let's take any other point P = (x, f (x)) on the graph, close to point A, and draw the secant line that passes through A and P. A secant line is a line that cuts the graph of a curve by one or more points.
To obtain the tangent line that we want, we only need to calculate the slope since we already have a point on the line: point A.
If we move point P along the graph and get it closer and closer to point A, the previously mentioned secant line will approach the tangent line that we want to find. Taking the limit when "P tends to A", both lines will coincide, therefore their slopes also.
The slope of the secant line is given by
Saying that P approaches A is equivalent to saying that "x" approaches "a". Thus, the slope of the tangent line to the graph of f at point A will be equal to:
The above expression is denoted by f '(a), and is defined as the derivative of a function f at the point “a”. We therefore see that analytically, the derivative of a function at a point is a limit, but geometrically, it is the slope of the line tangent to the graph of the function at the point.
Now we will look at this notion from the point of view of physics. We will arrive at the same expression of the previous limit, although by a different path, thus obtaining the unanimity of the definition.
The derivative as the instantaneous velocity of a moving object
Let's look at a brief example of what instantaneous velocity means. When it is said, for example, that a car to reach a destination did so with a speed of 100 km per hour, which means that in an hour it traveled 100 km.
This does not necessarily mean that during the whole hour the car was always 100 km, the car's speedometer could in some moments mark less or more. If you had the need to stop at a traffic light, your speed at that time was 0 km. However, after an hour, the journey was 100 km.
This is what is known as average speed and is given by the quotient of the distance traveled and the time elapsed, as we have just seen. Instantaneous speed, on the other hand, is the one that marks the needle of a car's speedometer at a given instant (time).
Let's look at this now more generally. Suppose that an object moves along a line and that this displacement is represented by the equation s = f (t), where the variable t measures time and the variable s the displacement, taking into account its beginning in the instant t = 0, at which time it is also zero, that is, f (0) = 0.
This function f (t) is known as the position function.
An expression is sought for the instantaneous velocity of the object at a fixed instant "a". At this speed we will denote it by V (a).
Let t be any instant close to instant "a". In the time interval between “a” and “t”, the change in the object's position is given by f (t) -f (a).
The average speed in this time interval is:
Which is an approximation of the instantaneous velocity V (a). This approximation will be better as t gets closer to "a". Thus, 
Note that this expression is the same as the one obtained in the previous case, but from a different perspective. This is what is known as the derivative of a function f at a point "a" and is denoted by f '(a), as stated above.
Note that making the change h = xa, we have that when "x" tends to "a", "h" tends to 0, and the previous limit is transformed (in an equivalent way) to:
Both expressions are equivalent but sometimes it is better to use one instead of the other, depending on the case.
The derivative of a function f at any point "x" belonging to its domain is then defined in a more general way as
The most common notation to represent the derivative of a function y = f (x) is the one we have just seen (f 'or y'). However, another widely used notation is Leibniz's notation which is represented as any of the following expressions:
Since the derivative is essentially a limit, it may or may not exist, since limits do not always exist. If it exists, the function in question is said to be differentiable at the given point.
Algebraic function
An algebraic function is a combination of polynomials by means of addition, subtraction, products, quotients, powers, and radicals.
A polynomial is an expression of the form
P n = a n x n + a n-1 x n-1 + a n-2 x n-2 +… + a 2 x 2 + a 1 x + a 0
Where n is a natural number and all a i, with i = 0,1,…, n, are rational numbers and n ≠ 0. In this case the degree of this polynomial is said to be n.
The following are examples of algebraic functions:
Exponential, logarithmic, and trigonometric functions are not included here. The derivation rules that we will see next are valid for functions in general, but we will restrict ourselves and apply them in the case of algebraic functions.
Bypass rules
Derivative of a constant
States that the derivative of a constant is zero. That is, if f (x) = c, then f '(x) = 0. For example, the derivative of the constant function 2 is equal to 0.
Derivative of a power
If f (x) = x n, then f '(x) = nx n-1. For example, the derivative of x 3 is 3x 2. As a consequence of this, it follows that the derivative of the identity function f (x) = x is f '(x) = 1x 1-1 = x 0 = 1.
Another example is the following: let f (x) = 1 / x 2, then f (x) = x -2 and f '(x) = - 2x -2-1 = -2x -3.
This property is also valid roots, since roots are rational powers and the above can also be applied in that case. For example, the derivative of a square root is given by
Derivative of addition and subtraction
If f and g are differentiable functions in x, then the sum f + g is also differentiable and it is satisfied that (f + g) '(x) = f' (x) + g '(x).
Similarly, we have that (fg) '(x) = f' (x) -g '(x). In other words, the derivative of a sum (subtraction), is the sum (or subtraction) of the derivatives.
Example
If h (x) = x 2 + x-1, then
h '(x) = (x 2) + (x)' - (1) '= 2x + 1-0 = 2x + 1.
Derived from a product
If f and g are differentiable functions in x, then the product fg is also differentiable in x and it is true that
(fg) '(x) = f' (x) g (x) + f (x) g '(x).
As a consequence, it follows that if c is a constant and f is a differentiable function in x, then cf is also differentiable in x and (cf) '(x) = cf' (X).
Example
If f (x) = 3x (x 2 +1), then
f '(x) = (3x)' (x 2 +1) + (3x) (x 2 +1) '= 3 (x)' (x 2 +1) + 3x
= 3 (1) (x 2 +1) + 3x = 3 (x 2 +1) + 3x (2x) = 3x 2 + 3 + 6x 2
= 9x 2 +3.
Derivative of a quotient
If f and g are differentiable at x and g (x) ≠ 0, then f / g is also differentiable at x, and it is true that
Example: if h (x) = x 3 / (x 2 -5x), then
h '(x) = / (x 5 -5x) 2 = / (x 5 -5x) 2.
Chain rule
This rule allows to derive the composition of functions. State the following: if y = f (u) is differentiable at u, yu = g (x) is differentiable at x, then the composite function f (g (x)) is differentiable at x, and it is true that '= f '(g (x)) g' (x).
That is, the derivative of a composite function is the product of the derivative of the external function (external derivative) and the derivative of the internal function (internal derivative).
Example
If f (x) = (x 4 -2x) 3, then
f '(x) = 3 (x 4 -2x) 2 (x 4 -2x)' = 3 (x 4 -2x) 2 (4x 3 -2).
There are also results for computing the derivative of the inverse of a function, as well as generalization to higher-order derivatives. The applications are extensive. Among them, its usefulness in optimization problems and maximum and minimum functions stand out.
References
- Alarcon, S., González, M., & Quintana, H. (2008). Diferential calculus. ITM.
- Cabrera, VM (1997). Calculation 4000. Editorial Progreso.
- Castaño, HF (2005). Mathematics prior to calculation. University of Medellin.
- Eduardo, NA (2003). Introduction to Calculus. Threshold Editions.
- Fuentes, A. (2016). BASIC MATH. An Introduction to Calculus. Lulu.com.
- Purcell, EJ, Rigdon, SE, & Varberg, DE (2007). Calculation. Pearson Education.
- Saenz, J. (2005). Differential Calculus (Second ed.). Barquisimeto: Hypotenuse.
- Thomas, GB, & Weir, MD (2006). Calculation: several variables. Pearson Education.