Contents
Daily Dose 024 | Mathematics
How do you find a Partial Derivative?
Define a Partial Derivative, what’s the point?
Welp, quite a bit.
We live in a three dimensional space where our designs need to operate safely and efficiently in the midst of a wide range of variables.
Of course we can’t design to maximize efficiency for every variable, but we can for the most critical.
Partial derivatives, loosely speaking, allow us to define how much a system (multi-variable function) will change when holding other variables constant.
If we stiffen the columns on a high rise at some point and increase the wind load, will the sway be limited to acceptable magnitude?
If we expanded the lifespan of this product by a year, will the economic return be worth the input?
So whether or not you see the value in working with Partial Derivatives in the real world, it’s something we need to know for the FE Exam.
We at Prepineer are going to make sure you are ready for them.
In this lesson, we dive in to a FE Exam Practice Problem in the subject of MATHEMATICS, specifically dealing with how to define a Partial Derivative.
Key Definition
What are PARTIAL DERIVATIVES?
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 to define a partial derivative?
Let’s talk partial derivatives, but more importantly how to define a partial derivative fast on your upcoming FE Exam.
First, let’s get a mental picture for what a partial derivative is.
Imagine cooking dinner for your family, where adjusting the amount of each ingredient affects the flavor profile. In mathematics and engineering, we deal with similar scenarios, but instead of spices and herbs, we might be looking at factors like the stress on a bridge or the flow of electricity in a circuit.
Partial derivatives are like measuring spoons that let us adjust one ingredient at a time in our recipe.
They show us how changing just one component, like salt or sugar, impacts the overall taste, while all other ingredients remain constant. It’s like fine-tuning the spiciness of a dish by altering just the chili content, without altering the quantity of other spices
In engineering, understanding these tiny changes is critical. It helps in designing everything from efficient engines to stable bridges.
You will see partial derivatives in a variety of forms on the FE Exam, here is a general process to solving them:
- Identify the Function
- Decide Which Variable to Differentiate With Respect to
- Apply the Partial Derivative Formula
- Use Standard Differentiation Rules
- Simplify the Result
- Repeat for Other Variables if Necessary
- Interpret the Result
Start with a multivariable function, such as f(x, y), which depends on two variables, x and y.
Choose whether to find the partial derivative with respect to x (keeping y constant) or with respect to y (keeping x constant).
If differentiating with respect to x, treat y as a constant and differentiate as you would a single-variable function. The reverse applies if differentiating with respect to y.
Apply standard differentiation rules, like the power rule, product rule, quotient rule, or chain rule, as appropriate. You can find all differentiation rules in the NCEES FE Reference Handbook.
Once you carry out the differentiation, simplify the expression to get the final form of the partial derivative.
If needed, repeat the process for other variables to find additional partial derivatives.
Understand that the resulting partial derivative represents the rate of change of the function in the direction of the variable you differentiated with respect to.
These problems can certainly present themselves more difficult than they are because of all the data and variable that you may be presented. However, following the step by step process will ensure that you are able to define a partial derivative fast, no matter how hard they initially appear.
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
