Partial derivatives

IntroductionPartial derivatives are a fundamental concept in multivariable calculus that is widely used in fields such as engineering, physics, and mathematics. A partial derivative is a derivative that is taken with respect to one variable, while all other variables are held constant. This is in contrast to a total derivative, which is the derivative with respect to all variables.
ExplanationThe mathematical definition of a partial derivative of a function, f(x,y), with respect to x, denoted as ∂f/∂x is given by:  
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∂f∂x=lim⁡h→0f(x+h,y)−f(x,y)h\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x,y)}{h}∂x∂f​=limh→0​hf(x+h,y)−f(x,y)​
where h is a small change in x. 
Similarly, the partial derivative formula of f(x,y) with respect to y, denoted as ∂f/∂y, is given by:
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∂f∂y=lim⁡h→0f(x,y+h)−f(x,y)h\frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x, y+h) - f(x,y)}{h}∂y∂f​=limh→0​hf(x,y+h)−f(x,y)​
Note that, while partial derivatives are taken with respect to a single variable, they are still a measure of how much the function f changes with a small change in x or y. These derivatives provide important information about the local behavior of a function and can be used to optimize, approximate, and solve complex problems in various fields.
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solve mathematical complex problems
The concept of partial derivatives can be generalized to higher dimensions by considering functions of more than two variables. For a function of n variables, f(x1, x2, ..., xn), the partial derivative with respect to x1 is given by:
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∂f∂x1=lim⁡h→0f(x1+h,x2....xn)−f(x1,x2...xn)h\frac{\partial f}{\partial x1} = \lim_{h \to 0} \frac{f(x1+h,  x2....xn) - f(x1, x2...xn)}{h}∂x1∂f​=limh→0​hf(x1+h,x2....xn)−f(x1,x2...xn)​
In addition to their theoretical importance, partial derivatives are widely used in engineering design and construction processes, such as structural analysis and optimization, material design, and fluid dynamics. They also play a crucial role in machine learning and artificial intelligence, where partial derivatives are used to optimize models and learn from data.
Application of partial differential equationPartial derivatives applications are found in various fields of science and engineering, including but not limited to:
Physics
Partial derivatives are used to describe the behaviour of physical systems, such as the velocity and acceleration of particles, the distribution of temperature, pressure or electric potential, and the flow of fluids and gases.
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Engineering
Partial derivatives are used in several engineering fields, such as civil, mechanical, aerospace and electrical engineering, to optimize designs and simulate real-world scenarios. For example, in civil engineering, partial derivatives are used to find the minimum resistance of a structure and to predict its stability. In mechanical engineering, partial derivatives are used to find the minimum power consumption of engines and to analyze the stress and deformation of materials.
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Economics
Partial derivatives are used in economics to study the behaviour of markets and to make predictions about supply and demand. For example, they are used to determine the effect of changes in interest rates on the price of a financial asset.
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Biology
Partial derivatives are used in various fields of biology, such as genetics, ecology and epidemiology, to model and understand the behaviour of populations and ecosystems.
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Machine Learning
Partial derivatives are used in machine learning to train neural networks and to optimize the parameters of algorithms. For example, the backpropagation algorithm uses partial derivatives to adjust the weights of a network and to minimize the error between the predicted and actual outputs.
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Image Processing
Partial derivatives are used in image processing to detect edges and features and to estimate the motion of objects in a video. For example, the gradient of an image can be used to detect lines and corners.
Partial derivatives are an essential tool for understanding and solving complex problems in a wide range of fields. They provide important information about the local behavior of functions and are used to optimize, approximate, and solve complex problems. Whether you are a university student or a STEM professional, a strong understanding of partial derivatives is critical for success in your field.
Additional ResourcesCheck out our articles on Vector Calculus, Integrals and Trigonometric Identities, and Differential Equations!
🔗 Vector Calculus and its Applications in Physics 
🔗 Integrals and Trigonometric Identities
🔗 Differential Equations and their Solutions
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