What is the derivative formula? What are the specific examples of the derivative formula under the 11th-grade Mathematics curriculum in Vietnam? Under the new program, what are the required outcomes regarding derivatives for 11th-grade students in Vietnam?

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What is the derivative formula? What are the specific examples of the derivative formula under the 11th-grade Mathematics curriculum in Vietnam?

11th-grade students may refer to the following information on the derivative formula and specific examples of the derivative formula under the 11th-grade Mathematics curriculum:

What is the derivative formula? What are the specific examples of the derivative formula under the 11th-grade Mathematics curriculum in Vietnam?

The derivative is an important concept in mathematics, especially in Calculus. It represents the rate of change of a function at a specific point on the graph of that function. In other words, the derivative indicates whether the function is increasing, decreasing, or constant at a certain point.

Geometric meaning: The derivative at a point is the slope of the tangent line to the graph of the function at that point.

*Example:

The graph of the function is increasing: Derivative is positive.

The graph of the function is decreasing: Derivative is negative.

The graph of the function reaches an extreme point (maximum or minimum): Derivative equals 0.

*Basic derivative formulas for 11th grade

Below are some basic derivative formulas that you often encounter in the 11th-grade Mathematics program:

*Derivative of basic functions:

Constant function: (c)' = 0 (where c is a constant)

Power function: (x^n)' = n*x^(n-1)

Exponential function: (a^x)' = a^x * ln(a)

Logarithmic function: (log_a(x))' = 1 / (x*ln(a))

*Trigonometric functions:

(sin(x))' = cos(x)

(cos(x))' = -sin(x)

(tan(x))' = 1/cos^2(x)

(cot(x))' = -1/sin^2(x)

*Derivative calculation rules:

Derivative of sum, difference: (u ± v)' = u' ± v'

Derivative of product: (u*v)' = u'v + uv'

Derivative of quotient: (u/v)' = (u'v - uv') / v^2 (with v ≠ 0)

Derivative of composite function: If y = f(u) and u = g(x) then y' = f'(u)*u'

Specific example

Problem: Calculate the derivative of the function y = 2x^3 + 5x - 1

Solution:

Apply the derivative formula of sum, difference, and power functions, we have:

y' = (2x^3)' + (5x)' - (1)'

y' = 23x^(3-1) + 5*1 - 0

y' = 6x^2 + 5

Thus, the derivative of the function y = 2x^3 + 5x - 1 is y' = 6x^2 + 5.

Problem: Calculate the derivative of the function y = sin(2x)

Solution:

This is an example of a composite function. Set u = 2x, then y = sin(u).

Apply the derivative formula of composite functions and the derivative of the sine function, we have:

y' = cos(u)*u'

y' = cos(2x)*2

y' = 2*cos(2x)

Thus, the derivative of the function y = sin(2x) is y' = 2*cos(2x).

*Note: Information is for reference only./.

What is the derivative formula? Provide specific examples of the derivative formula for 11th grade? What requirements should 11th-grade students meet according to the new program?

Under the new program, what are the required outcomes regarding derivatives for 11th-grade students in Vietnam?

Under subsection 3, Section 5 of the General Education Program in Mathematics issued under Circular 32/2018/TT-BG/DDT, the required outcomes regarding derivatives for 11th-grade students in Vietnam include:

Content	Required outcomes
Concept of derivative. Geometric meaning of the derivative	- Recognize some problems leading to the concept of a derivative such as: determining the instantaneous velocity of an unevenly moving object, determining the rate of temperature change. - Recognize the definition of a derivative. Be able to calculate the derivative of some simple functions by definition. - Recognize the geometric meaning of the derivative. - Establish the equation of the tangent line of the function graph at a point on the graph. - Recognize the number e through problems modeling bank interest rates.
Rules for calculating derivatives	- Be able to calculate the derivative of some basic elementary functions (such as polynomial functions, simple radical functions, trigonometric functions, exponential functions, logarithmic functions). - Be able to use formulas for calculating the derivative of the sum, difference, product, quotient of functions and derivative of composite functions. - Solve some problems related to other subjects or practical issues associated with derivatives (e.g., determining the instantaneous velocity of an unevenly moving object,...).
Second derivative	- Recognize the concept of the second derivative of a function. - Be able to calculate the second derivative of some simple functions. - Solve some problems related to other subjects or practical issues associated with the second derivative (e.g., determining acceleration from the velocity-time graph of an uneven motion,...).

What are the objectives of the 11th-grade Mathematics curriculum in Vietnam regarding basic mathematical knowledge and skills?

Under the General Education Program in Mathematics issued under Circular 32/2018/TT-BGDĐT, specific regulations are as follows:

Mathematics at the lower secondary level aims to help students achieve the following main objectives:

+ Contribute to the formation and development of mathematical competence with the requirement to: pose and answer questions during reasoning, solve problems, perform reasonable reasoning when solving problems, and prove not-too-complex mathematical propositions;

+ Be able to use mathematical models (mathematical formulas, algebraic equations, representations,...) to describe situations in some uncomplicated real-life problems; use mathematical language combined with ordinary language to express mathematical content as well as demonstrate evidence, reasoning, and results; present ideas and how to use tools and means for learning mathematics to perform a learning task or to describe arguments or proofs in mathematics.

+ Have basic mathematical knowledge and skills about:

++ Numbers and Algebra: The number system (from natural numbers to real numbers); calculation and use of calculation tools; algebraic language and symbols; transform algebraic expressions, equations, systems of equations, inequalities; use function language to describe (model) some processes and phenomena in reality.

++ Geometry and Measurement: Geometry and Measurement content at this education level include Visual Geometry and Plane Geometry.

Visual Geometry continues to provide language, symbols, and descriptions (at a visual level) of real-world objects (plane shapes, solid shapes); establish some common geometric models; calculate some geometric elements; develop spatial imagination; solve some simple practical problems related to Geometry and Measurement.

Plane Geometry provides knowledge and skills (at a reasoned logic level) about geometric relationships and some common plane figures (points, lines, rays, line segments, angles, two parallel lines, triangles, quadrilaterals, and circles).

++ Statistics and Probability: Collect, classify, represent, analyze, and process statistical data; analyze statistical data through frequency, and relative frequency; recognize some simple statistical laws in practice; use statistics to understand the basic concepts of the experimental probability of an event and the probability of an event; recognize the meaning of probability in practice.

+ Help students have an initial understanding of careers related to mathematics; have a sense of career orientation based on abilities, interests, conditions, and personal circumstances; and orient streamlining after lower secondary education (continue studying, vocational training, or engage in working life).

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