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Daily Dose 039 | Mathematics
How to uncomplicate solving partial derivatives?
Solving partial derivatives doesn’t need to be complicated.
With a very basic understanding of a framework, these types of problems can become ones that you will be hoping you see come the day of your exam.
In this video, we break it all down, showing you exactly how to go about solving partial derivatives faster and with a lot less stress on the FE Exam.
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
So how can we simplify solving Partial Derivatives?
Oftentimes, solving partial derivatives can seem daunting, but they don’t need to be.
Given the problem statement:
Determine the partial derivative with respect to y of the function x^2 + 3xy + y^2.
We will walk one by one through the steps needed to make solving partial derivatives quick and easy.
- Step 1: Determine the variable with respect to which you want to take the partial derivative.
- Step 2: Apply the derivative operator.
- Step 3: Differentiate each term.
- Step 4: Combine the derivatives.
- Step 5: Simplify if possible.
In our example, the problem statement specifically states that we are to find the partial derivative with respect to y.
Next we apply the partial derivative operator (∂/∂x) to the function with respect to the chosen variable, in this case y.
We will treat all other variables as constants.
With the variable defined, differentiate each term of the function with respect to the chosen variable, following the standard differentiation rules.
Differentiating the first term:
∂/∂y (x^2) = 0 (since x^2 is independent of y)
Differentiating the second term:
∂/∂y (3xy) = 3x
Differentiating the third term:
∂/∂y (y^2) = 2y
Combine the derivatives of each term to obtain the final partial derivative expression.
Doing so gives us the expression ∂f/∂y = 0 + 3x + 2y = 3x + 2y.
Lastly, we need to smplify the expression if possible by combining like terms or using any available algebraic simplifications.
Wrapping up our example, we find that our expression is simplified as ∂f/∂y = 3x + 2y
Check out the video and see how we can go about solving this type of problem in the most efficient manner.
As always, with Love, Prepineer
