The first thing to understand, in order to perform the derivative of a function, is its meaning. In general terms, a function can be represented as a machine, in which there are some unknowns that, depending on the value that we enter, will give us another value, such as:
F(x)= x^2
- If we substitute x=1, it will return us, as value y=1,
- if we enter x=4, it will return y= 16 and vice versa.
All these values are represented in a graph, which is composed of the ordinate axis (Y), and the abscissa axis, (X). Once this is understood, we can more easily understand the definition of a derivative.
We can define the derivative of a function as: The slope of the tangent line at an exact point on the graph, that is, the study of the variation of the graph, at each of its exact points.
Once all of the above is understood, it is necessary to know the steps to follow, for each of the existing cases, in which the derivatives can be applied.
To calculate the slope of a curve, it would be enough to use some drawing techniques. For example, in the curve that makes this drawing [Curve «C»]. It’s a line drawn on a very popular diagram, known as the “Cartesian coordinate system,” where the horizontal axis is called the “abscissa” and the vertical axis is called the “ordinate.” If we use a Cartesian system to paint our curve, we will have more facilities to calculate the specific location of each point on the curved line.
Derivation Table Explanation.
- Derivative of a constant is 0, and the identity function is 1.
- Derivative of a potential function with n belonging to the real ones.
- The derivative of a simple potential function is equal to the exponent multiplied by its base minus one.
- The derivative of a simple irrational function, 1 divided by the index of the root, by the root itself, with radicand n-1.
- The derivative of a simple irrational fraction is equal to the inverse expression of the product of the index, by the root of the same index, of the power n-1 of the radicand x.
- The derivative of an exponential function is equal to the same function, times the natural logarithm of its base.
- The derivative of a logarithmic function is equal to 1, divided by the product of the unknown, of the logarithmic function, by the natural logarithm of its base.
- The derivative of the potential exponential function is equal to the sum of the derivatives of an exponential function and a potential function.
- The derivative of sine x is always cosine x.
- The derivative of the cosine is the – sine.
- The derivative of the tangent function is equal to 1, plus the square of the tangent.
- The derivative of the arcsine function is equal to 1 divided by the square root of 1 – x squared.
- The derivative of the arc cosine function is equal to the derivative of the arc sine function, but negated.
- The arc tangent function is equal to 1 times 1 plus x squared.
For composite functions, they could be summarized as, the steps cited above, multiplied by the derivative of the function that contains it.
It is important to study this table, in order to apply the use of derivatives, depending on the type of function we have. If they do not know the formulas of each derivative, they will not be able to do any type of exercise or problem that contains them. In the event that they are not known by heart, it is recommended that the table be printed.
Operations with derivatives: (D= Derivative)
- Sum: D(f(x) + g (x)): D (f(x)) +D (g(x)) or also said the sum of two functions is equal to the particular sum of each function.
- Subtraction: D(f(x) – g (x)) =: D (f(x)) -D (g(x)) to easily summarize the subtraction of two functions is the same as performing the particular subtraction of each one of the functions.
- Function multiplication: D(f(x)*g(x)) =: D (f(x)) * g(x) + f(x) * D g(x) that is, to perform the derivative of a multiplication you must derive the first function and multiply it by the second without differentiation added by the first function without differentiation multiplied by the second derivative function.
- Division of functions: (D (f(x)) * g(x) – f(x) * D g(x)) g(x)^2. In other words, the derivative of the first function must be performed by the second non-derivative function subtracted by the first non-derivative function multiplied by the second derived function and all of this divided by the second function squared.
Function properties.
- Multiplication of a constant K by a function: D (K*f(x)) = K* D (f(x))
- Chain rule: D (f(g(x)) = D (f(g(x)) * D (g(x)) (It is simply based on deriving the first function and therefore multiplying it by the function that compose)
Once the basic theory for understanding and calculating derivatives has been learned and understood, we will carry out some simple exercises for your better understanding when practicing:
Exercises and Problems
Exercise number 1:
D [e- ²×] = e – ²× * -2.
Explanation: This function is of the compound exponential type, so we must leave the same function, multiplied by the derivative of the function that it has in its index, which is minus two, since the derivative of an identity function is one. Therefore, negative two times one is negative two.
D [ Ln ( 3ײ) * ( 5׳ -7)³] = D [ Ln ( 3ײ)] + D [3Ln( 5׳ -7)] = 6x/ 3ײ + 3 * 15 ײ / 5׳ -7 = 2 / x + 45 ײ / 5׳ -7 =
10׳ – 14 + 45׳ / x (5׳ -7 )= 55׳ – 14 / x (5׳ -7 )
/ = The bar of the fraction, it is not a division.
Explanation: Before doing this exercise, you should have some basic knowledge about the properties of logarithms:
- The multiplication of logarithms D [ Ln ( 3ײ) * ( 5׳ -7)³] is equal to the sum of the two functions that contain it.
- D[ Ln(3ײ)] + D[3Ln( 5׳ -7)]
- The division of logarithms D [ Ln ( 3ײ )/ (5׳ – 7 )³)] is equal to the subtraction of the two functions that contain it.
- D[ Ln(3ײ)] – D[3Ln( 5׳ -7)]
- Any logarithm of exponent” x”, for example, In (5׳ -7) ³ is equal to the exponent, multiplied by the logarithm . 3 Ln ( 5׳ -7)
- The logarithm of 0 does not exist, and the logarithm of 1 is 0.
- The logarithm to the base of a of a, is equal to 1.
- The logarithm to the base of n of a to the power of n is equal to n.
- The logarithm of negative numbers does not exist, since the domain of a logarithmic function is included for x > 0.
- Once these rules are known, the resulting fractions are operated, to make the derivatives of the logarithms of each of the functions, and we simplify the result obtaining: 60 ׳ -14 / x ( 5׳ -7).
Example number 2:
D [sin ( 2x) + cos (ײ)] = cos (2x )* 2 – sin (ײ) * 2x.
Explanation: The derivative of a sum as explained above is done through the partial derivatives of the functions being added. We will start by deriving sin ( 2x), which, as we know, being a function composed of a sine function, will give us the result, by the derivative of the function that contains 2x, obtaining 2, since we are performing the derivative of an identity function. Then we will proceed to derive cos (ײ), which as we know, the derivative of a function composed of a cosine, is equal to – sin (ײ), by the function it contains, which in this case we find a potential function, thus obtaining 2x cos (2x) * 2- sin (ײ) * 2x.
Example number 3:
/ = The bar of the fraction, it is not a division.
D (arc tc 2x) = 2x * In 2 / 1 + 2² x
Explanation: Because it is a function composed of an arctangent function, according to the formula for the derivative of the arctangent, it would be obtained 1 / 1 + 2² x multiplied by the derivative of the function that contains it, in the case of an exponential function .2x * In2
Problem number 1:
Find the equations of the lines tangent and normal to the curve y = 2׳ + x at the origin of coordinates.
F(x) = +2׳x.
- The equation of the tangent line is: y –y1 = m ( x –×1 ).
Explanation: Because the problem asks for the equations of the lines located at the origin of coordinates, the value of x = 0, therefore, we will substitute f (x) for f (0), obtaining the value 0. Once After this step, the derivative of f (0) must be operated, which is 6 * 0² + 1 = 1.
Tangent line: y – f(0) = f´(0) * (x-0) = f´(x) =6ײ + 1 = f´(0) =1
Y – 0 = 1 (x-0) = y=x.
- The equation of the normal line is: y1 – = 1/D y1 ( x –x1 ).
Explanation: As explained above, the same steps will be followed, except for the slope m, which is what we indicate as -1 / f´ (0), operating as we have done previously, we would obtain as a result:
Y – 0 = -1 (x – 0) = y = -x.
Normal line: yf(0)= -1 / fx (0) * (x -0)
/ = The bar of the fraction, it is not a division.
Problem number 2
Find the values of a and b for which the tangent line to the curve y=ײ + ax + b at the point P (3,0) has slope 2.
The graph of the given curve passes through the point P (3,0) = 0 =9 +3a+b.
Because they give us the points, x= 3 and y= 0, we must substitute them in the given function, obtaining as a result the one indicated below, and we will also perform the derivative of it, in order to be able to find the value of a, and therefore the value of b:
F´ (3) = 2 = as f´(x) =2x + a = f´ (3) = 6 + a = 6 + a= 2 = a = -4
Therefore, a = -4 AND b = -9 – 3rd = 3 = a= -4 and b =3