MAT 21C Lecture Notes - Lecture 20: Partial Derivative, Heat Capacity
MAT 21C – Lecture 20 – Partial Derivatives for Multi-Variable Functions
• Partial Derivatives: Suppose we have a function of two variables, f(x, y). Then the
partial derivative of f with respect to x annotated as: or
is the derivative
that treats y as a constant (y is held fixed). The partial derivative with respect to
y annotated as: or
is the y-derivative with x held fixed.
• Example: If , find and .
To find, , we treat as constants. Then
. Following similar logic,
Here, we treat as constants.
• Formal Definition:
and
.
• Example: Let
.
Suppose . Then it implies that for any non-zero value of x and non-zero
value of y,
What about
and
?
Suppose, we set only y = 0. Then
Since
for any value of x, then it implies that
. Similarly, if we set x = 0,
then
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Mat 21c lecture 20 partial derivatives for multi-variable functions: partial derivatives: suppose we have a function of two variables, f(x, y). Then the that treats y as a constant (y is held fixed). To find, (cid:1858)(cid:3051)(cid:4666)(cid:1876),(cid:1877)(cid:4667), we treat (cid:1857)(cid:3052),(cid:1877),(cid:1856) (cid:1877)(cid:2871) as constants. Here, we treat sin(cid:4666)(cid:1876)(cid:4667)(cid:1856) (cid:1876)(cid:2870) as constants: formal definition: (cid:1858)(cid:3051)(cid:4666)(cid:1876)0,(cid:1877)0(cid:4667)= lim 0(cid:4666)(cid:3051)0+ , (cid:3052)0(cid:4667) (cid:4666)(cid:3051)0,(cid:3052)0(cid:4667) and (cid:1858)(cid:3052)(cid:4666)(cid:1876)0,(cid:1877)0(cid:4667)= lim 0(cid:4666)(cid:3051)0, (cid:3052)0+(cid:4667) (cid:4666)(cid:3051)0,(cid:3052)0(cid:4667, example: let (cid:1858)(cid:4666)(cid:1876),(cid:1877)(cid:4667)= {(cid:882),(cid:1876)(cid:1877) (cid:882) (cid:883),(cid:1876)(cid:1877)=(cid:882). Then it implies that for any non-zero value of x and non-zero value of y, (cid:3051)(cid:4666)(cid:1876),(cid:1877)(cid:4667)= (cid:3052)(cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:882). Since (cid:3051)(cid:4666)(cid:1876),(cid:882)(cid:4667)= (cid:882) for any value of x, then it implies that (cid:3051)(cid:4666)(cid:882),(cid:882)(cid:4667)=(cid:882). We know that (cid:1858)(cid:4666)(cid:1876),(cid:882)(cid:4667)=(cid:883) and (cid:1858)(cid:4666)(cid:1876),(cid:1876)(cid:4667)=(cid:882) (cid:1875) (cid:1876) (cid:882). Sometimes, we write (cid:4666)(cid:3051)(cid:4667)(cid:3052) partial derivative with x and y fixed. derivatives with two variables. The equation for its state is represented by = (cid:4666),(cid:4667). The second derivative of (cid:1858)(cid:3051) with (cid:1858)(cid:3051)= (cid:885)(cid:1876)(cid:2870)(cid:1877)+(cid:1857)(cid:3051)sin (cid:4666)(cid:1877)(cid:4667) respect to y is (cid:1858)(cid:3051)(cid:3052)=(cid:885)(cid:1876)(cid:2870)+(cid:1857)(cid:3051)cos (cid:4666)(cid:1877)(cid:4667) (cid:1858)(cid:3052)=(cid:1876)(cid:2871)+(cid:1857)(cid:3051)cos (cid:4666)(cid:1877)(cid:4667)
