dy/dx vs dx/dy: Difference and Comparison

Key Takeaways

1. The notation ‘dy/dx’ represents the derivative of a function y concerning x.
2. The notation ‘dx/dy’ represents the derivative of the inverse function x concerning y.
3. Dy/dx is commonly used in physics, engineering and natural sciences, whereas dx/dy is used in applications that involve economics and optimization problems.

What is dy/dx?

The notation ‘dy/dx’ represents the derivative of a function y concerning x. It is a fundamental concept in calculus that describes how the output of a function changes as its input varies. Essentially, it measures the rate of change of y concerning modifications. This concept is crucial in understanding the behaviour of functions and is used extensively in various fields of science and engineering.

When we write dy/dx, we ask: How much does the output y change for a slight change in the input x? The derivative provides an instantaneous rate of change at a specific point on the function’s curve. Geometrically, it corresponds to the slope of the tangent line to the curve at that point. Calculating dy/dx involves using differentiation rules and techniques such as the power, chain, and product rules.

What is dx/dy?

‘dx/dy’ represents the derivative of the inverse function x concerning y. While less commonly used, it has its significance. It is helpful when dealing with parts where expressing inverse relationships is more accessible.

In essence, dx/dy allows us to understand how small changes in y affect changes in x for an inverse function. Calculating dx/dy involves applying differentiation techniques to the inverse function. This is useful in solving certain differential equations, where expressing the relationship in terms of the inverse function makes the calculations more manageable.

To grasp the concept of dx/dy, envision a graph where x is the horizontal axis and y is the vertical axis. The slope of the tangent line at a specific point on the graph corresponds to the value of dx/dy at that point.

Difference Between dy/dx and dx/dy

1. The dy/dx notation represents how the rate of change of y varies as x changes, whereas the dx/dy note means how the rate of change of x varies as y changes.
2. Dy/dx is the standard notation used extensively in calculus and differential equations, whereas dx/dy is less commonly used and appears in specialized contexts like implicit differentiation.
3. In the graphical representation of dy/dx, a tangent line is drawn to a curve at a point representing the instantaneous rate of change of y concerning x. In contrast, for dx/dy, a tangent line is drawn to a turn at an end, representing the instantaneous rate of change of x concerning y.
4. Dy/dx is commonly used in physics, engineering and natural sciences, whereas dx/dy is used in applications that involve economics and optimization problems.
5. It is easier to interpret dy/dx in chain rule application, whereas the result expression is hard to solve with dx/dy.

Comparison Between dy/dx and dx/dy

Parameters	dy/dx	dx/dy
Notation meaning	Represents how the rate of change of y varies as x changes	Describes how the rate of change of x varies as y changes
Usage	Notation is used extensively in calculus and differential equations	It is less commonly used and appears in specialized contexts like implicit differentiation
Graphical interpretation	A tangent line is drawn to a curve at a point representing the instantaneous rate of change of y concerning x	The tangent line is drawn to a curve at a point representing the instantaneous rate of change of x concerning y
Applications	Physics, engineering and natural sciences	Economics and optimization problems
Chain rule application	Easier to interpret	Harder to interpret

References

1. https://wrap.warwick.ac.uk/502/1/WRAP_Tall_dot1991j-visual-dif-ft-mt.pdf
2. https://link.springer.com/chapter/10.1007/978-1-4757-3949-7_1