Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus ↓


Differential Calculus


Differential calculus is a branch of calculus that studies how functions change when the inputs change. It is mainly concerned with the concept of the derivative, which is a way of measuring the rate of change of a function at a given point. In simple terms, differential calculus helps us understand how to find the slope of a curve at a specific point, which can be thought of as the instantaneous rate of change.

Introduction to derivatives

The derivative of a function is a fundamental concept in differential calculus. It is denoted as f'(x) or frac{df}{dx}, and it represents the rate of change of the function f(x) with respect to the variable x. The derivative can be thought of as the slope of the tangent line to a curve at a point.

Finding derivatives

To understand how to find derivatives, consider a function f(x) that describes a curve. Suppose we want to find the slope of the curve at a particular point x = a. The derivative at this point corresponds to the slope of the tangent line at this point. Mathematically, derivatives are defined as limits:

lim_{{h to 0}} frac{f(a + h) - f(a)}{h}

This formula shows the slope of the secant line that approaches the tangent line when h approaches zero.

Basic rules of differentiation

The process of finding a derivative is called differentiation. Several basic rules simplify differentiation for many functions:

Power rule

The power rule states that if f(x) = x^n, then the derivative f'(x) = n cdot x^{n-1}. For example, the derivative of x^3 is 3x^2.

Continuation rule

If f(x) is a constant c, then the derivative f'(x) = 0 In this case, the function does not change, so its rate of change is zero.

Sum and difference rules

If f(x) and g(x) are functions, then the derivative of their sum (or difference) is

(f pm g)'(x) = f'(x) pm g'(x)

This means that you can identify each part separately and then add or subtract the results.

Product rule

For two functions f(x) and g(x), the derivative of their product is given by:

(f cdot g)'(x) = f'(x) cdot g(x) + f(x) cdot g'(x)

Quotient rule

For two functions f(x) and g(x), the derivative of their quotient is given by:

left(frac{f}{g}right)'(x) = frac{f'(x) cdot g(x) - f(x) cdot g'(x)}{g(x)^2}

Visualization of derivatives

To understand derivatives better, consider the graph of a function. At any point on this graph, the derivative tells us the slope of the tangent line, which indicates the rate of change of the function at that point. Here is an example SVG illustration of a tangent line to a curve:

In this visualization, the blue curve represents the function, and the red line is the tangent at a specific point where it touches the curve. The slope of this red line represents the derivative value at that point on the function.

Applications of differential calculus

Differential calculus is incredibly useful in a variety of fields, including physics, engineering, economics, and biology. It helps us model and solve real-world problems where rates of change are involved. Here are some examples:

Physics

In physics, the concepts of velocity and acceleration are derivatives. Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. Using differential calculus, scientists can calculate the instantaneous velocity and acceleration of objects in motion.

Economics

In economics, differential calculus is used to find marginal cost and marginal revenue, which are derivatives of the cost and revenue functions, respectively. This allows economists to determine the additional cost or revenue incurred by producing one more unit of a commodity.

Biology

Biologists use differential calculus to model population dynamics. The rate at which populations of organisms grow or decline is often modeled using derivatives.

Text example

Let's illustrate how to use differential calculus by solving some basic derivative calculations:

Example 1: Find the derivative of a polynomial

Consider the function f(x) = 3x^4 - 5x^2 + 2x - 7 Using the power and sum rules, the derivative is:

f'(x) = 12x^3 - 10x + 2

Example 2: Find the derivative of a product

Let u(x) = 2x^3 and v(x) = 3x - 4 Using the multiplication rule:

(u cdot v)' = u' cdot v + u cdot v'

The derivatives of u and v are u'(x) = 6x^2 and v'(x) = 3 Applying the multiplication rule gives:

(2x^3 cdot (3x-4))' = 6x^2 cdot (3x-4) + 2x^3 cdot 3
= 18x^3 - 24x^2 + 6x^3
= 24x^3 - 24x^2

Example 3: Derivative of a quotient

Consider f(x) = frac{x^2}{x + 1} Using the quotient rule:

left(frac{x^2}{x+1}right)' = frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2}

Simplifying this gives:

frac{2x(x+1) - x^2}{(x+1)^2}
= frac{2x^2 + 2x - x^2}{(x+1)^2}
= frac{x^2 + 2x}{(x+1)^2}

Conclusion

Differential calculus is an essential tool in mathematics that helps us understand and calculate the rates at which changes occur. By exploring derivatives and their rules, we gain insight into real-world phenomena and can solve complex problems in a variety of scientific disciplines. This powerful branch of calculus opens the door to deeper analysis in science, economics, engineering, and beyond.


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