Differentiation of the difference

Differentiation

The algebraic process of calculating derivatives is known as differentiation. The slope or gradient of a particular graph at any given position is the derivative of a function. The tangent’s value drawn to that curve at any given location is the gradient of that curve. The curve’s gradient varies at different positions along the axis for non-linear curves. As a result, calculating the gradient in such situations is challenging.

It’s also known as a property’s change about another property’s unit change.

Consider the function f(x) as a function of the independent variable x. The independent variable x is caused by a tiny change in the independent variable Δx. The function f(x) undergoes a similar modification Δf(x). Ratio  Δf(x)/Δx is a measure of f(x) of change in relation to x.

As Δx approaches zero, the ratio’s limit value is

limΔx tends to 0f(x+Δx)-f(x)Δx  is known as the first derivative of the function f(x).

Derivative

One of the most powerful concepts in mathematics is the derivative of a function. A function’s derivative is usually a new function known as the derivative function or the rate function.

A function’s derivative is defined as the rate at which the value of the dependent variable changes in relation to the change in the value of the independent variable. It’s a primary calculus tool that can also be translated as the slope of a tangent line. It determines how steep a function’s graph is at a given location on the graph.

The rate at which a function changes at a specific point is known as the derivative.

Differential

Differential calculus, along with integral calculus, is one of the most fundamental divisions of calculus. It’s a calculus branch that deals with tiny changes in varying quantities. Unfortunately, this world is full of interconnected amounts that fluctuate regularly.

For example, a circular body’s area changes as the radius changes or the velocity of a projectile changes. In mathematics, these changing entities are called variables, and the rate of change of one variable about another is referred to as a derivative. The connection between these variables is expressed by an equation known as a differential equation.

Relationship between Differential and Derivative

Differentiation is a technique for calculating a derivative, the rate of change of a function’s output y based on variable x.

Simply said, derivative refers to the rate of y with respect to x, and this connection is represented as y = f(x), implying that y is a function of x. The function that determines the slope’s value of f(x) where it is specified and f(x) is differentiable is called the derivative of the function f(x). It’s the graph’s slope at a specific position.

Function

A function is a relationship between a group of inputs with one output. A function, in simple terms, is a relationship between inputs in which each input is associated with only one output. Each function has a domain and a co-domain, often known as a range. The general notation is f(x), where x represents the input. A function’s general representation is y = f(x).

In mathematics, a function is a specific relationship between inputs, the domain, and outputs, the co-domain. Each information has exactly one word, which can be traced back to its input.

Linear and Nonlinear Functions

A nonlinear function is easily defined as a function that is not linear. As a result, understanding a linear function is necessary before understanding a nonlinear function. A linear function has a line as its graph. In algebra, a linear function is a polynomial with the largest exponent equal to 1 or a horizontal line (y = c, c is a constant here). A function with a continuous slope is also known as a linear function (rate of y with respect to x). That means the slope of the line connecting any two locations on the function is equal.

Let’s define a nonlinear function which is already known. Non-linear functions, as previously established, are functions that are not linear. As a result, they exhibit the polar opposite characteristics of a linear function. A line is the graph of a linear function. As a result, the graph of a nonlinear function isn’t a straight line. Non-linear functions have a slope that varies across points, whereas linear functions have a constant slope. Linear functions are polynomials with the highest exponent of 1 or of the form y = c where c is constant in algebra. All other functions are nonlinear functions.

Differentiation of the difference of the function

The derivative of a function which is the difference of two other functions is equal to the difference of their derivatives, which is given. This can be demonstrated using the first principle technique or the derivative by definition method.

Take a function, y=f(x)-g(x)

Step 1: Take increment in the function:

y+Δy=f(x+Δx)-g(x+Δx)

Δy=f(x+Δx)-g(x+Δx)-y

Step 2: Put value of y from Step 1 in above equation

Δy=f(x+Δx)-g(x+Δx)-f(x)+g(x)

Δy=f(x+Δx)-f(x)-g(x+Δx)+g(x)

Δy=[f(x+Δx)-f(x)]-[g(x+Δx)-g(x)]

Step 3: Divide both sides by Δx

Δy/Δx=[f(x+Δx)-f(x)]-[g(x+Δx)-g(x)]/Δx

Δy/Δx=[f(x+Δx)-f(x)]/Δx – [g(x+Δx)-g(x)]/Δx

Step 4: Take limit of both sides as Δx →0

limΔx →0Δy/Δx=limΔx →0[[f(x+Δx)-f(x)]/Δx – [g(x+Δx)-g(x)]/Δx]

limΔx →0Δy/Δx=limΔx →0[f(x+Δx)-f(x)]/Δx – limΔx →0[g(x+Δx)-g(x)]/Δx

dy/dx=LimΔx →0[f(x+Δx)-f(x)]/Δx -lim Δx →0[g(x+Δx)-g(x)]/Δx

This will give

dy/dx=f’(x)-g’(x)

This demonstrates that the difference of the derivatives of the two functions is equal to the difference of their derivatives.

Conclusion

In this article, we gained knowledge about differentiation, differential and derivative. We also found out about their relationship. Functions were also introduced, and we also read about linear and nonlinear functions. In the end, we grasped how to find differentiation of the difference.