Partial Derivatives:
The partial derivative of the function of numerous variables is its derivative with respect to one of those variable with others held constant.
x = x(z,y)........................................Eq.1
From equation 1, x a dependent variable is the function of two independent variables z and y . Partial derivative of x with respect to y with z held constant is (∂x/∂y)z
For example, if
x = zy2........................................Eq.2
Then, partial derivative of x with respect to y with z held constant is
(∂x/∂y)z = 2zy........................................Eq.3
Likewise, partial derivative of x with respect to z with y held constant is
(∂x/∂z)y = y2........................................Eq.4
Exact Differential:
Assume that there exists the relation among three coordinates x, y, and z in such a way that x is a function of y and z (i.e. x(z, y) ); therefore
f(x, y, z) = 0........................................Eq.5
Exact differential of x (dx) is
dx(∂x/∂y)z dy + (∂x/∂z)ydz........................................Eq.6
Usually for any three variables x , y , and z we have relation of form
dx = M(y,z)dy + N(y,z)dz........................................Eq.7
If differential dx is exact, then
(∂M/∂z)y = (∂N/∂y)z........................................Eq.8
Implicit Differential:
Consider the equation of form
xy = x2y2........................................Eq.9
One can differentiate two sides of equation 9 using equation 6 (i.e differentiating both the left and right hands side with respect to x while y is held constant and with respect to y while x is held constant).
(∂(xy)/∂x)ydx + (∂(xy)/∂y)xdy = (∂(x2y2)/∂x)ydx + (∂(x2y2)/∂y)xdy........................................Eq.10
Equation 10 gives
ydx + xdy = 2xy2dx + 2x2ydy........................................Eq.11
Collecting like term and then factorize to have
dy/dx = (2xy2 - y)/(x - 2x2y) ........................................Eq.12
Consider equation 9 as f = xy - x2 y2 (i.e. moving expression in right side of equation 9 to left side and then equate result to f ). Then
dy/dx = -(∂f/∂x)/(∂f/∂y) ........................................Eq.13
Product of Three Partial Derivatives:
Assume that there exists the relation among three coordinates x, y, and z; therefore
f(x, y, z) = 0........................................Eq.14
Then x can be imagined as the function of y and z
dx = (∂x/∂y)zdy + (∂x/∂z)ydz........................................Eq.15
Also y can be imagined as the function of x and z, and
dy = (∂y/∂x)zdx + (∂y/∂z)xdz........................................Eq.16
Insert equation 16 in 15
dx = (∂x/∂y)z[(∂y/∂x)zdx + (∂y/∂z)xdz] + (∂x/∂z)ydz
Rearrange to get:
dx = (∂x/∂y)z(∂y/∂x)zdx + [(∂x/∂y)z(∂y/∂z)x + (∂x/∂z)y]dz........................................Eq.17
If dz = 0 dx ≠ 0 it follows that
(∂x/∂y)z(∂y/∂x)z = 1
(∂x/∂y)z = 1/((∂y/∂x)z)........................................Eq.18
In eq.17 if dx = 0 and dz ≠ 0, it follows that:
(∂x/∂y)z(∂y/∂z)x + (∂x/∂z)y = 0
Move (∂x/∂z)y to other side of equation to get
(∂x/∂y)z(∂y/∂z)x = -(∂x/∂z)y........................................Eq.19
Then divide both sides of equation 19 by (∂z/∂x)y
(∂x/∂y)z(∂y/∂z)x(∂z/∂x)y = -1........................................Eq.20
This is known as minus-one product rule.
Chain Rule of Partial Derivatives:
Another helpful relation is known as chain rule of partial derivatives. Assume T is function of V and P, and that each of V and P is the function of Z, then
(∂T/∂V)P = (∂T/∂Z)P(∂Z/∂V)P........................................Eq.21
Equation 21 is chain rule of partial derivative. The following can as well be written:
(∂S/∂P)T = (∂S/∂V)T(∂V/∂P)T........................................Eq.22(a)
(∂U/∂V)P = (∂U/∂T)P(∂T/∂V)P........................................Eq.22(b)
Equation 21 and 22 are known as chain rule of partial derivatives
Second Derivatives or Second Order Derivatives:
Let f (x, y) be function with continuous order derivatives, then we can compute first derivatives to be (∂f/∂x)z and (∂f/∂z)x. One can further compute second derivatives ∂2f/∂x2, ∂2f/∂z2 ∂2f/∂x∂y, and ∂2f/∂y∂x. Take note of these two second order derivates i.e. ∂2f/∂x∂y and ∂2f/∂y∂x, they are known as mixed second derivatives. It can be shown that mixed second derivatives are equal, i.e. it doesn't matter order will perform differentiation.
∂2f/∂x∂y = ∂2f/∂y∂x
Functions of More than Two Variables:
Assume that f (x, y, z), derivative of f with respect to one of the variables with other two constant (e.g. derivative of f x with y and z constant) can be written as:
(∂f/∂x)yz, (∂f/∂y)xz, and (∂f/∂z)xy
Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)
Expand your confidence, grow study skills and improve your grades.
Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.
Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.
Free to know our price and packages for online physics tutoring. Chat with us or submit request at [email protected]
Synthesis and Reactions of Benzofuran and Benzothiophene tutorial all along with the key concepts of Physical and Chemical Properties, Synthesis of Benzofuran and Benzothiophenes
Life cycle and classification of bryophytes tutorial all along with the key concepts of Features of Bryophytes, General life cycle, Morphology of Bryophytes, Hepaticopsida and Anthocerotopsida
Online STAR Exam Preparation course and online STAR tutoring package offered by www.tutorsglobe.com are the most comprehensive and customized collection of study resources on the web, offering best collection of STAR Practice papers, quizzes, STAR test papers, and guidance.
Qualitative analysis of common functional group tutorial all along with the key concepts of Functional group identification tests, General Scheme of Analysis, Solubility classification, Solubility classification, Tests for unsaturation, Functional group classification tests
Accounting for by-products - By-products are together generated products of minor significance and do not contain separate costs until the split off point.
www.tutorsglobe.com offers system used in dfd homework help, assignment help, case study, writing homework help, online tutoring assistance by computer science tutors.
the basic gates are - not gate, and gate, or gate, nand gate, nor gate, xor gate, xnor gate.
tutorsglobe.com dynamics of filtration assignment help-homework help by online mechanism of urine formation tutors
tutorsglobe.com artificial intelligence assignment help-homework help by online computer science tutors
Chromosome Theory of Inheritance tutorial all along with the key concepts of Deduction of Chromosome Theory of Inheritance, Comparison of Development in Two Dispermic Embryos, Support of chromosome Theory
Theory and lecture notes of Graphs and Equations all along with the key concepts of graphs and equations, homework help, assignment help and understanding macroeconomics. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Graphs and Equations.
tutorsglobe.com cuttings assignment help-homework help by online artificial technique of vegetative propagation tutors
the power can be calculated basically with the help of just ammeter and voltmeter.
homologous series-functional groups and isomerism tutorial all along with the key concepts of organic compounds by functional groups, structural isomerism, geometric isomerism
physical and chemical characteristics of alkali metals tutorial all along with the key concepts of alkali metals occurrence, extraction of alkali metals, uses of alkali metals, melting points-boiling points, thermal-electrical conductivity, ionisation energy, ionic character of compounds, lattice energy-hydration energy
1933780
Questions Asked
3689
Tutors
1490454
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!