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


Differentiation is a fundamental concept in calculus that deals with the idea of change. It is widely used in engineering, physics, economics, and other fields. At its core, differentiation allows us to figure out the rate at which one quantity changes relative to another. In simple terms, it helps us understand how something changes over time or space.

Understanding slopes and rates of change

To understand differentiation, let's start with the concept of slope. Imagine a straight line on a graph. The slope of this line tells us how steep it is. Mathematically, slope is defined as the ratio of the "rise" (change in the vertical direction) and the "run" (change in the horizontal direction).

slope = rise / run = (change in y) / (change in x)

Now, let's imagine this:

Point A Point B

In the above example, the line goes through points A and B. The slope of the line is calculated by taking the change in the y-values and dividing it by the change in the x-values.

When dealing with curves instead of straight lines, the concept of slope becomes the concept of tangent. The tangent to a curve at any point is a straight line that "touches" the curve at that point. The slope of the tangent line at a particular point represents the rate of change of the curve at that specific point.

Definition of derivative

The derivative of a function represents the rate of change of the output value of a function with respect to its input value. In other words, it is the slope of the tangent line to the function at a given point. The derivative of a function is often represented as f'(x) or df/dx, where f is the original function.

The derivative is defined mathematically as:

f'(x) = lim (h → 0) [f(x + h) - f(x)] / h

This expression is known as the "difference quotient." It takes the limit of the average rate of change of the function over the interval [x, x + h] as h approaches 0.

Computing derivatives

Let's calculate the derivatives of some basic functions to understand these concepts better.

Derivative of a constant function

Consider a constant function f(x) = c, where c is a constant. The graph of this function is a horizontal line. It does not change, so the rate of change is zero. Therefore, the derivative of a constant function is:

f'(x) = 0

Derivative of a linear function

Now consider a linear function f(x) = mx + b. The graph of this function is a line with slope m. As expected, the derivative of a line is its slope:

f'(x) = m

Derivative of a power function

Let's find the derivative of the power function f(x) = x^n, where n is a real number. Using the power rule, which states that the derivative of x raised to any power n is:

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

For example, if f(x) = x^2, then

f'(x) = 2x

If f(x) = x^3, then

f'(x) = 3x^2

Visualization of derivatives

Consider the function f(x) = x^2. The derivative of this function is f'(x) = 2x. This means that as x changes, the slope of the tangent line to the curve changes. Let's visualize this:

Tangent at x=1 f(x) = x^2

The curve represents the function f(x) = x^2. The red line is the tangent to the curve at a point near x=1. Note that the slope of this tangent is steeper at higher values of x, as described by the derivative f'(x) = 2x.

Higher order derivatives

The process of differentiation can be repeated. The derivative of a function f(x) is a new function that can itself be differentiated. The second derivative of f(x), denoted as f''(x), provides information about how the rate of change of the original function is changing.

For example, consider f(x) = x^3:

f(x) = x^3
f'(x) = 3x^2
f''(x) = 6x

The second derivative helps us understand the "curvature" of the original function. It answers the question, "Is the rate of change increasing or decreasing?"

Practical applications of differentiation

Differentiation has many applications in the real world. Here are some examples:

1. Physics

In physics, differentiation is used to calculate velocity and acceleration. If the position of an object is given by a function relative to time, then the derivative of this function with respect to time gives the velocity of the object. The second derivative gives the acceleration of the object.

2. Economics

In economics, differentiation can be used to find marginal cost or revenue. Marginal cost is the cost of producing one more unit of a good, and it can be calculated using the derivative of the cost function.

M'(x) = dC/dx

3. Biology

In biology, differentiation is used to model population growth rates and other dynamic systems.

Conclusion

Differentiation is a powerful mathematical tool that helps us understand and quantify change. From analyzing the slope of a straight line to modeling complex dynamic systems in a variety of scientific fields, differentiation provides essential insights into the world around us. By practicing calculating derivatives and visualizing their meanings, we gain a deeper understanding of both mathematical theory and practical applications.


Grade 11 → 4.2


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