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Daily Dose 018 | Mathematics
How do you find the Partial Derivative of a function?
I’ve noticed a consistent trend when one is first asked to find the partial derivative of a function.
All our students in Prepineer start with a heavy emphasis on the foundational concepts of Mathematics – everything that we encounter in the practice of engineering relies on our ability to confidently operate within this realm.
However, while most start off strong in the area of College Algebra, most start to hit a wall at the point of Differential Equations.
But it doesn’t need to be this way, in this video, we dive into an FE Exam Practice Problem in the subject of Mathematics, specifically revolving around Partial Derivatives.
Key Definition
What is a PARTIAL DERIVATIVE?
In MATHEMATICS, the PARTIAL DERIVATIVE of a function of several variables is its derivative with respect to one of those variables, with the others held constant.
This is opposed to the total DERIVATIVE, in which all variables are allowed to vary.
The reason for a new type of derivative is that when the input of a function is made up of multiple variables, we want to see how the function changes as we let just one of those variables change while holding all the others constant.
The Process
What is the process of finding the Partial Derivative?
Being asked to find the partial derivative of a function may present itself as difficult, and maybe even a bit unuseful, but it’s really not hard – and it’s very relevant and useful in the world of engineering.
A partial derivative allows us to determine how a system is affected when a certain variable changes, while keeping all others constant.
As an example, in Heat Transfer Analysis, partial derivatives are used to describe how temperature changes in a material as a function of time and space, which is essential in designing heating and cooling systems.
Other engineering applications include, but are not limited to:
- Stress Analysis: In materials and structures, partial derivatives help determine how stress varies with force and moment, crucial for ensuring structural integrity.
- Signal Processing: Partial derivatives are applied in signal processing to understand how signals change over time and space, impacting the design of filters and circuits.
- Fluid Mechanics: In fluid flow problems, partial derivatives describe how pressure, velocity, and other properties of the fluid change, which is fundamental in water resource management and infrastructure design.
- Aerodynamics: Partial derivatives help in calculating changes in pressure and temperature over aircraft surfaces, influencing lift and drag forces and overall aircraft performance.
Knowing how to work with and find the partial derivative of a function is crucial not only in Mathematics, but in real-world engineering practice.
So we’ve got to ensure we are comfortable with the process, because we are going to see them on the FE Exam.
The following procedure gives you a general framework to find the partial derivative of a function fast and efficiently no matter how complex of a scenario you are presented.
Let’s run through the steps:
- Identify the Function
- Select the Variable of Interest
- Apply Differentiation Rules
- Simplify the Expression
- Repeat for Other Variables (if needed)
- Interpret the Results
- Review Your Work
Start with a multivariable function, typically expressed as f(x, y, z, …), where x, y, z, … represent the variables.
Choose one variable with respect to which you wish to take the partial derivative. Consider all other variables as constants during this step.
Use the standard rules of differentiation (power rule, product rule, quotient rule, chain rule) to differentiate the function with respect to the chosen variable.
After taking the derivative, simplify the result as much as possible to arrive at the partial derivative expression.
If you need to find partial derivatives with respect to other variables, repeat the steps for each variable.
The partial derivative obtained gives you the rate of change of the function with respect to the chosen variable, while other variables are held constant.
Check your calculations to ensure accuracy. Reviewing helps catch any possible mistakes in applying differentiation rules or simplifying the expression.
Following these general steps will ensure you can confidently find the Partial Derivative of a function no matter the engineering application.
And with all of that being stated, check out the video and see how we can go about solving this type of problem in the most efficient manner.
As always, we are here to help, Prepineer
