Partial Derivative
Rate of change with respect to one variable, holding others fixed.
Formula
Variables
Example
Treats y as constant
How it works
A partial derivative measures how a multivariable function changes when you vary just one input and freeze all the others. It is built from the same limiting idea as an ordinary derivative, but the limit is taken along a line parallel to the chosen axis, so only one variable is allowed to move. This makes it the basic tool for isolating the contribution of one variable in systems where several inputs interact.
When to use it
Use it whenever a quantity depends on more than one variable and you need the sensitivity to just one of them, such as in optimization, thermodynamics, economics, or machine learning. It is especially useful when analyzing local behavior of surfaces, gradients, and directional changes in high-dimensional models.
Worked example, step by step
Let f(x,y)=x^2y+3y^4-5x. We want ∂f/∂x, so treat y as constant and differentiate with respect to x. ∂/∂x(x^2y)=y·∂/∂x(x^2)=2xy ∂/∂x(3y^4)=0 because y is constant ∂/∂x(-5x)=-5 So, ∂f/∂x=2xy-5 At (x,y)=(4,-2): ∂f/∂x = 2(4)(-2)-5 = -16-5 = -21 Final answer: ∂f/∂x(4,-2) = -21
Common mistakes to avoid
- Differentiating every variable instead of holding the others constant
- Forgetting that terms made only of other variables become constants and differentiate to 0
- Confusing a partial derivative with a total derivative when the variables are linked by another relation
Did You Know?
Partial derivatives, holding all but one variable fixed, are the language of thermodynamics and machine learning.
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