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Daily Dose 029 | Mathematics
How do you solve complex partial differentiation?
Partial Differentiation is one of those topics that tends to fall in the realm of uncertainty.
In some cases, these problems are easy to solve.
In other, more complex cases, we don’t even know where to start.
‘Tis the reality of the FE Exam.
Though we often feel prepared enough to see anything, the NCEES always has its ways of throwing us off our game.
In this lesson, we dive into an FE Exam Practice Problem, this time within the subject of MATHEMATICS, specifically revolving around PARTIAL DIFFERENTIATION of a complex function.
Key Definition
What is PARTIAL DIFFERENTIATION?
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
How to solve complex partial differentiation problems fast?
Partial differentiation is a crucial tool needed by engineers and mathematicians – and certainly on the FE Exam.
Let’s start with a mental picture of what partial differentiation is.
Imagine you’re in a park with paths leading in all directions.
Deciding to take a step in any one direction affects your view and position, but what if you could measure exactly how much that step changes your perspective?
This is what partial differentiation allows us to do in the Engineering mathematical landscape.
In our world, we often work with functions that depend on more than one variable. For instance, the temperature inside a room could change based on its height off the ground and its distance from a window.
Partial differentiation helps us understand how changing one variable, while keeping others constant, affects the whole system.
It’s like focusing on how moving closer to or farther from the window changes the temperature, without going up or down at all.
So what steps will you take when asked to solve complex partial differentiation problem on the FE Exam?
Follow this step by step process to make a seemingly complex problem into something that is routine:
- Step 1: Understand the Function
Identify and isolate the multivariable function you are working with. This function should depend on two or more variables, for example, f(x, y).
- Step 2: Choose the Variable for Differentiation
Decide which variable you will partially differentiate with respect to, holding the others constant. This could be x, y, or another variable within your function.
- Step 3: Apply the Partial Differentiation Rule
Use the rules of differentiation to find the partial derivative with respect to your chosen variable. Remember, once you decide which variable you will be differentiating, you will treat all the other variables as constants.
- Step 4: Simplify Your Result
Simplify the result of your differentiation to get the partial derivative. This might involve basic algebraic manipulation.
- Step 5: Repeat for Other Variables
If the problem requires finding partial derivatives with respect to multiple variables, repeat the process for each variable.
- Step 6: Interpret Your Results
Consider what the partial derivative is telling you about the function’s behavior in relation to each variable. This can give insights into rates of change along different dimensions.
- Step 7: Apply to Engineering Problems
Use your calculated partial derivatives to solve the engineering problem at hand, whether it’s optimizing a design or analyzing a system’s behavior under variable conditions.
Adhering to these steps will guide you in solving even the most complex partial differentiation problems.
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
