Calculus is one of the most dreaded yet the most important branches of Mathematics. It is centred on two major concepts, integrals and derivatives. Most students do not understand why they have to go through the torture of mastering the two concepts and their applicability; hence the question, what is dx? This list of calculus terms and their meanings deciphers the concepts and their significance.
Mathematically, calculus is used to obtain optimal solutions. Therefore, it is used to decipher the changes between values related by a function. While doing so, it focuses on topics and concepts such as integration, differentiation and limit functions.
What Is dy/dx
Before taking the deep dive to understand what dy/dx means, it is important to highlight that calculus is divided into two main sections:
- Integration calculus
- Differential calculus
Both integral and differential calculus provide a foundation for a complex branch of Mathematics called Analysis. They deal with the impact of a slight change in a dependent variable as it approaches zero on the function. So, what is a function?
A Function
In calculus, a function represents the relationship between two variables; a dependent and independent variable. This equation explains what a function is:
y=f(x)
In this expression, x is the independent variable, f is the function, and y is the dependent variable.
Differential Calculus
Differential calculus solves the problem by finding the rate of change of a function with reference to the other variables. The derivatives are used to calculate the maximum and minimum values of a function to find the optimal solution.
What is DX differentiation? Differential calculus helps determine the limit of the quotient by dealing with variables such as x and y with respect to the function. Therefore, dx and dx are referred to as differentials. The process of determining the derivatives is known as differentiation.
The derivative of a function y with respect to x is expressed as dy/dx or f'(x).
Limits
Limits are used to calculate the degree of closeness to any value. The limit formula is used to show a limit, and it is expressed as:
limx→cf(x) = A
Grammatically, the expression is read as “the limit of f of x as x approaches c equals A”.
Derivatives
In calculus, derivatives are used to represent instantaneous rates of change of a quantity with respect to another. Therefore, the derivative of a function is expressed as:
limx→h[f(x + h) − f(x)]/h = A
Continuity
A function, f(x) is considered continuous if x=a if these three conditions are fulfilled:
f(a) is defined
limx→af(x) exists
limx→a− f(x) = limx→a+ f(x) = f(a)
Continuity and Differentiability
A function is considered continuous if it is differentiable at any point. If not, it is not continuous.
Integral Calculus
Integral calculus focuses on the study of integrals and their associated properties.
Integration
In layman's language, integration is the reciprocal of differentiation. In simpler terms, differentiation involves dividing a section into smaller parts, while integration involves collecting the small sections to form a larger section.
Definite Integral
A definite integral has a specified boundary for the calculation of the function. This means that the upper and lower limits of the independent variable are specified. Mathematically, a definite integral is expressed as:
∫ab f(x).dx = F(x)
Indefinite Integral
Unlike the definite integral, an indefinite integral has no boundaries. Therefore the integration value is always accompanied by (C), regarded as a constant value. An indefinite integral is expressed as:
∫ f(x).dx = F(x) + C
Calculus Formulae
Cumulatively, Calculus formulae are divided into six broad sets of formulae. The six formulae are integration, differentiation, integrals, application of differentiation, and differential equations. These formulae complement each other.
Limit Formulae
Limit formulae help in approximating a value to a defined number. They are defined by either zero or infinity. Limit formulae include:
- Ltx→0 (xn - an)(x - a) = na(n - 1)
- Ltx→0 (sin x)/x = 1
- Ltx→0 (tan x)/x = 1
- Ltx→0 (ex - 1)/x = 1
- Ltx→0 (ax - 1)/x = logea
- Ltx→0 (1 + (1/x))x = e
- Ltx→0 (1 + x)1/x = e
- Ltx→0 (1 + (a/x))x = ea
Differentiation Formulae
Differentiation formulae apply to trigonometric ratios, inverse trigonometry, basic algebra and exponential terms.
Definite Integrals Formulae
Definite integrals are integral formulae with defined upper and lower limits. Definite integrals formulae include:
- ∫ba f'(x).dx = f(b) - f(a)
- ∫ba f(x).dx = ∫ba f(t).dt
- ∫ba f(x).dx = - ∫ab f(x).dx
- ∫ba f(x).dx = ∫ca f(x).dx + ∫bc f(x).dx
- ∫ba f(x).dx = ∫ba f(a + b - x).dx
- ∫a0 f(x).dx = ∫a0 f(a - x).dx
- ∫2a0 f(x).dx = 2∫a0 f(x).dx
- ∫a-a f(x).dx = 2∫a0 f(x).dx, f is an even function
- ∫a-a f(x).dx = 0 , f is an odd function
Application of Differentiation Formulae
The application of differential formulae is used to estimate values and approximation. The formulae are also used to find the maxima and minima, equations of tangents and normals, and the changes of numerous physical events.
Commonly used differential formulae include:
- dy/dx = (dy/dt)/(dx/dt)
- Equation of a Tangent: y - y1 = dy/dx.(x - x1)
- Equation of a Normal: y - y1 = -1/(dy/dx).(x - x1)
Differential Equations Formula
Differential equations that have higher-order derivatives are comparable to general equations. In general, we have an unknown variable "x" and the variable of the equation as the differentiation of dy/dx.
- Common differential equations formulae include:
- Homogeneous Differential Equation: f(λx, λy) = λnf(x, y)
- Linear Differential Equation: dy/dx + Py = Q
- General solution of Linear Differential Equation is y.e- ∫P.dx = ∫(Q.e∫P.dx).dx + C
What Is dx in Medical Terms?
Forget about d(x) in calculus; it denotes the diagnosis in medical terms. It determines the nature of the disease.
What Does dx Mean in Business?
In business, dx refers to digital transformation. It is a strategy that enables business innovation to be predicted by incorporating digital technology.
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