Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus → Differentiation ↓


Definition of Derivative


In the field of calculus, one of the fundamental concepts students encounter is the idea of a "derivative." Derivatives play a key role in understanding how functions change, describing rates of change, and solving many real-world problems. As we explore this topic, we will delve deeper into the definition of the derivative, examine its mathematical basis, understand through visual examples, and look at practical examples.

What is a derivative?

The derivative shows the rate at which a function is changing at a given point. If you imagine a graph representing a function, the derivative at a particular point tells you how steep the graph is at that point, or how quickly the values are rising or falling. In simple terms, if you think of a journey, the derivative tells you at what speed you are traveling.

Mathematical definition

The derivative of a function f(x) at a specific point x is defined as the limit of the average rate of change of the function over an interval, as the interval shrinks to zero. Mathematically, it can be expressed as:

lim[h -> 0] (f(x + h) - f(x)) / h

In this equation:

  • lim denotes the limit.
  • h is a small increase in x.
  • f(x + h) is the value of the function at x + h.
  • f(x) is the value of the function at x.

For example, if f(x) = x^2, then the derivative, f'(x), is calculated as:

        f'(x) = lim[h -> 0] ((x + h)^2 - x^2) / h 
              = lim[h -> 0] (x^2 + 2xh + h^2 - x^2) / h 
              = lim[h -> 0] (2xh + h^2) / h 
              = lim[h -> 0] (2x + h) 
              = 2x

Understanding the derivative visually

Let us try to understand graphically what the derivative means. Imagine that you are looking at the graph of the function f(x). The derivative at a point on the graph represents the slope of the tangent line to the curve at that point.

tangent line

In the above picture:

  • The red dot is a point on the curve of the function f(x).
  • The blue line is the tangent at that point, representing the derivative.
  • The green dashed line is a secant line used in calculating the derivative.

The slope of the blue line is the derivative. If the line slopes upward, the derivative is positive; if it slopes downward, the derivative is negative.

Other examples

Example 1: Linear function

Consider f(x) = 3x + 5 To find the derivative:

        f'(x) = lim[h -> 0] ((3(x + h) + 5) - (3x + 5)) / h 
              = lim[h -> 0] (3h) / h 
              = 3

The derivative of a linear function is the constant slope of the line, which is 3 in this case.

Example 2: Polynomial function

Consider f(x) = x^3. To find the derivative:

        f'(x) = lim[h -> 0] ((x + h)^3 - x^3) / h 
              = lim[h -> 0] (x^3 + 3x^2h + 3xh^2 + h^3 - x^3) / h 
              = lim[h -> 0] (3x^2h + 3xh^2 + h^3) / h 
              = lim[h -> 0] (3x^2 + 3xh + h^2) 
              = 3x^2

The derivative of x^3 is 3x^2, which shows how the slope of the tangent changes with x.

Example 3: Power function

The derivative rule for any power function f(x) = x^n is:

        f'(x) = n*x^(n-1)

This formula, known as the power rule, is a simple and efficient way to find derivatives of power functions.

Example 4: Trigonometric functions

Consider f(x) = sin(x). The derivative is obtained using trigonometric limits and identities:

        f'(x) = cos(x)

This means that the rate of change of the sine function is represented by the cosine function.

Applications of derivatives

Derivatives have many applications in various sectors. Here are some of the major applications:

  • Physics: Derivatives help in understanding speed, velocity, and acceleration.
  • Economics: They are used in economic models to determine marginal costs and revenues.
  • Biology: Used in modeling population growth and decay.
  • Engineering: Essential for understanding changes and optimization in various processes.

Conclusion

Understanding derivatives is a cornerstone of calculus, opening the door to many mathematical applications and insights. Remember that derivatives are simply a tool for measuring how things change. While derivatives can be complex, breaking them down into their basic concepts, as we have done here, can make them manageable and accessible.

As you continue your studies in calculus, you will find that derivatives are just the beginning. They form the basis for integral calculus, differential equations, and much more. Keep practicing and using these concepts, as they will serve as valuable tools in your mathematical toolbox.


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Definition of Derivative

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