Section 13.2
Partial Derivatives
What if we only change one variable at a time?
1
Introduction
For a function , we can ask: "How does change if we move only in the -direction?"
This is the Partial Derivative with respect to , denoted or .
The Rule:
To find , treat as a constant and differentiate normally with respect to .
2
Geometric Interpretation
Interactive: Slopes on a Surface
is just the slope of the surface in the -direction (holding constant).
3
Worked Examples
Example 1: Polynomials
Find and for .
1. Find
Treat as constant.
.
(It's a constant!).
So, .
2. Find
Treat as constant.
.
.
So, .
Example 2: Product Rule
Find for .
Use Chain Rule. Outer function is , inner is .
(Remember is constant coeff here).
.
Example 3: Quotient Rule (Advanced)
Find for .
Recall Quotient Rule: , taking derivs wrt .
4
Practice Quiz
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