When learning calculus, we encounter terms like Dx/Dy and Dy/Dx quite often. These two terms are essential to understanding the differentials, derivatives, and partial derivatives of functions. In this article, we will dive into the details of Dx/Dy Vs Dy/Dx and explore how they are related to calculus.
What is Dx/Dy?
Dx/Dy refers to the derivative of ‘x’ with respect to ‘y.’ In simple terms, it is the change in ‘x’ with respect to the change in ‘y.’ Dx/Dy is often used when dealing with implicit functions.
Implicit functions are those that cannot be written in the form of an explicit equation. Instead, they are represented as an equation between the variables x and y. We can find the derivative of an implicit function by taking the derivative of both sides of its equation with respect to ‘y.’
For example, consider the implicit function,
x^2 + xy – y^3 = 0
To find the derivative of ‘x’ with respect to ‘y,’ we can differentiate both sides of the equation as follows:
2x(dx/dy) + x(dy/dy) + y(dx/dy) – 3y^2(dy/dy) = 0
Simplifying this equation, we get:
(dx/dy) = -(x+ y(dy/dx))/ (2x- 3y^2)
In this case, we have used the quotient rule to find the derivative of x with respect to y.
What is Dy/Dx?
Dy/Dx refers to the derivative of ‘y’ with respect to ‘x.’ In other words, it is the change in ‘y’ with respect to the change in ‘x.’ Dy/Dx is commonly used when dealing with explicit functions.
An explicit function is one that can be written in the form of a simple equation, such as y = f(x). We can find the derivative of an explicit function by taking the derivative of the equation with respect to ‘x.’
For example, consider the function,
y = x^3 – 3x^2 + 2x + 1
To find the derivative of y with respect to x, we can differentiate the equation as follows:
dy/dx = 3x^2 – 6x + 2
In this case, we have applied the power rule to find the derivative of x^3 and the constant rule to find the derivative of 1. We have also used the sum rule to add up the derivatives of all the terms in the expression.
Dx/Dy Vs Dy/Dx: How are they related?
Dx/Dy and Dy/Dx are related through the chain rule of differentiation. The chain rule is a fundamental rule of calculus that tells us how to differentiate compositions of functions. It applies to both explicit and implicit functions.
The chain rule states that if y is a function of u, and u is a function of x, then:
(dy/dx) = (dy/du) * (du/dx)
Similarly, if x is a function of v, and v is a function of y, then:
(dx/dy) = (dx/dv) * (dv/dy)
These two equations are simply different ways of stating the same chain rule in terms of different variables. To obtain one equation from the other, we can invert both sides of the second equation and substitute u = v^(-1):
(dv/dy) = (1/ (dx/dy))
(dx/dy) = (1/ (dv/dy))
Thus, we can see that Dx/Dy and Dy/Dx are inverse of each other, i.e., if we are given one, we can find the other using the inverse function principle.
In practice, the choice between Dx/Dy and Dy/Dx depends on the context of the problem. If we are dealing with an implicit function, we may use Dx/Dy, while explicit functions may require the use of Dy/Dx.
Dx/Dy Vs Dy/Dx is an essential concept in calculus, enabling us to understand the differentials, derivatives, and partial derivatives of functions. These two terms are related by the chain rule, which applies to both implicit and explicit functions. In general, we choose Dx/Dy for implicit functions and Dy/Dx for explicit functions. This knowledge proves valuable when solving calculus problems and can be applied to real-life situations, paving the way for innovative solutions.