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- Difference Quotient
- Partial Derivative Symbol
- Partial Derivative Examples
- Partial Derivative Rules
- Second Partial Derivative
- Mixed Partial Derivative
- Partial Derivative Chain Rule
- Partial Derivative Product Rule
- Find The Indicated Partial Derivative
- Inflection Point
- Linear Approximation
- Newton Raphson Method

The function of an independent variable say y = f(x). Here, we learn about quantities which depend on more than one independent variable.

For example Area of a rectangle depends on its length(x) and width(y).

which can be expressed as A = f(x,y)

Volume of a parallel piped depends on its length l, width w and height h.

Hence Volume = f(l, w, h)

Hence “Partial Differentiation” deals with the differentiation of a function of many independent variables.

For example Area of a rectangle depends on its length(x) and width(y).

which can be expressed as A = f(x,y)

Volume of a parallel piped depends on its length l, width w and height h.

Hence Volume = f(l, w, h)

Hence “Partial Differentiation” deals with the differentiation of a function of many independent variables.

i.e u = f( x, y )

Let us define the partial derivative of f with respect to x and y separately as follows, which is similar to the

$\frac{\partial u}{\partial x}$

$\frac{\partial u}{\partial y}$

[ $\partial$() is the symbol of partial derivative usually read as “del” and it should not be written as ‘$\delta$ ‘ (delta) ]

The derivative of u with respect to x is called the partial derivative of u with respect to x,

and is denoted by u

The derivative of u with respect to y is called the partial derivative of u with respect to y,

and is denoted by u

f(x,y) = sin(x

Given f(x,y) = sin(x

Now, finding out f

f

when we keep y as constant cos y becomes a constant so its derivative becomes zero.

Similarly, finding f

fy = $\frac{\partial f}{\partial y}$ = cos(x2 y2) $\times$ 2 y x

f(x , y) = xy + x

Given f(x , y)=xy + x

Let's, finding out f

f

Now, for the same problem we try to find out partial derivative with respect to y

f

f

Let u = f( x, y).

We are aware of the first order partial derivatives , u

The second order partial derivatives will be, uxx = $\frac{\partial}{\partial x}$ $(\frac{\partial u}{\partial x} )$ = $\frac{\partial^{2} u}{\partial x^{2}}$

u

u

u

It is very important to note that, $\frac{\partial^{2} u}{\partial y \partial x}$ = $\frac{\partial^{2} u}{\partial x \partial y}$

u

if x = x(t) and y = y(t)

then $\frac{dx}{dt}$ = x ' (t) and $\frac{dy}{dt}$ = y ' (t)

The partial derivatives will be, u

then,

$\frac{du}{dx}$ = ( 3x + 5y ) $\frac{d}{dx}$ (2x

= ( 3x + 5y ) ( 2. 2x) + ( 2x

= 4x ( 3x + 5y ) + 3 (2x

R = ln $(u^{2} + v^{2} + w^{2})$

u = x + 2y,

v = 7x - y,

w = 3xy; when x = y = 4.

Given R = ln $ (u^{2} + v^{2} + w^{2}) $

u = x + 2y

v = 7x - y

w = 3xy

When x = y = 4:

u = 4 + 2 $\times$ 4 = 12

v = 7 $\times$ 4 - 4 = 24

w = 3 $\times$ 4 $\times$ 4 = 48

$\frac{\partial R}{\partial x}$ = ($\frac{\partial R}{\partial u}$ $\times$ $\frac{\partial u}{\partial x}$) + ($\frac{\partial R}{\partial v}$ $\times$ $\frac{\partial v}{\partial x}$) + ($\frac{\partial R}{\partial w}$ $\times$ $\frac{\partial w}{\partial x}$)

$\frac{\partial R}{\partial x}$ = $\frac{2u}{(u^{2}+v^{2}+w^{2})(1)}$ + $\frac{2v}{(u^{2}+v^{2}+w^{2})(7)}$ + $\frac{2w}{(u^{2}+v^{2}+w^{2})(3y)}$

$\frac{\partial R}{\partial x}$ = $\frac{2 \ \times \ 12}{(12^{2} \ + \ 24^{2} \ + 48^{2})(1)}$ + $\frac{2 \ \times \ 24}{(12^{2} \ + \ 24^{2} \ + 48^{2})(7)}$ + $\frac{2 \ \times \ 48}{(12^{2} \ + \ 24^{2} \ + 48^{2})(3 \ \times \ 4)}$

$\frac{\partial R}{\partial x}$ = $\frac{24}{3024 \times 1}$ + $\frac{48}{3024 \times 7}$ + $\frac{96}{3024 \times 12}$

$\frac{\partial R}{\partial x}$ = $\frac{24}{3024}$ + $\frac{336}{3024}$ + $\frac{1152}{3024}$

$\frac{\partial R}{\partial x}$ = $\frac{1512}{3024}$

$\frac{\partial R}{\partial x}$ = $\frac{1}{2}$

$\frac{\partial R}{\partial y}$ = $(\frac{\partial R}{\partial u} \times \frac{\partial u}{\partial y}) + (\frac{\partial R}{\partial v} \times \frac{\partial v}{\partial y}) + (\frac{\partial R}{\partial w} \times \frac{\partial w}{\partial y})$

$\frac{\partial R}{\partial y}$ = $\frac{2u}{(u^{2}+v^{2}+w^{2})(2)}$ + $\frac{2v}{(u^{2}+v^{2}+w^{2})(-1)}$ + $\frac{2w}{(u^{2}+v^{2}+w^{2})(3x)}$

$\frac{\partial R}{\partial y}$ = $\frac{2 \ \times \ 12}{(12^{2} \ + \ 24^{2} \ + 48^{2})(2)}$ + $\frac{2 \ \times \ 24}{(12^{2} \ + \ 24^{2} \ + 48^{2})(-1)}$ + $\frac{2 \ \times \ 48}{(12^{2} \ + \ 24^{2} \ + 48^{2})(3 \ \times \ 4)}$

$\frac{\partial R}{\partial y}$ = $\frac{24}{3024 \times 2}$ + $\frac{48}{3024 \times (-1)}$ + $\frac{96}{3024 \times 12}$

$\frac{\partial R}{\partial y}$ = $\frac{48}{3024}$ - $\frac{48}{3024}$ + $\frac{1152}{3024}$

$\frac{\partial R}{\partial y}$ = $\frac{1152}{3024}$

$\frac{\partial R}{\partial y}$ = $\frac{8}{21}$

Y = w tan

u = r + s,

v = s + t,

w = t + r;

$\frac{\partial Y}{\partial r}$,

$\frac{\partial Y}{\partial s}$,

$\frac{\partial Y}{\partial t}$

when r = 0, s = 1, t = 2

Given Y = w tan

u = r + s,

v = s + t,

w = t + r;

$\frac{\partial Y}{\partial r}$ = $(\frac{\partial Y}{\partial u})(\frac{\partial u}{\partial r}) + (\frac{\partial Y}{\partial v})(\frac{\partial v}{\partial r}) + (\frac{\partial Y}{\partial w})(\frac{\partial w}{\partial r})$

$\frac{\partial Y}{\partial r}$ = $(\frac{w \times v}{(1 + (uv)^{2})})$(1) + $(\frac{w \times u}{(1 + (uv)^{2})})$(0) + (tan-1(uv))(1)

$\frac{\partial Y}{\partial r}$ = $\frac{w \times v}{(1 + (uv)^{2})} + tan

$\frac{\partial Y}{\partial s}$ = $(\frac{\partial Y}{\partial u})(\frac{\partial u}{\partial s}) + (\frac{\partial Y}{\partial v})(\frac{\partial v}{\partial s}) + (\frac{\partial Y}{\partial w})(\frac{\partial w}{\partial s}) $

$\frac{\partial Y}{\partial s}$ = $(\frac{w \times v}{(1 + (uv)^{2})})$ (1) + $(\frac{w \times u}{(1 + (uv)^{2})})$ (1) + (tan

$\frac{\partial Y}{\partial s}$ = $\frac{w \times v}{(1 + (uv)^{2})} + \frac{w \times u}{(1 + (uv)^{2})})$

$\frac{\partial Y}{\partial s}$ = $\frac{w(v + u)}{1 + (uv)^{2}} $

$\frac{\partial Y}{\partial t}$ = $(\frac{\partial Y}{\partial u})(\frac{\partial y}{\partial t})$ + $(\frac{\partial Y}{\partial v})(\frac{\partial v}{\partial t})$ + $(\frac{\partial Y}{\partial w})(\frac{\partial w}{\partial t})$

$\frac{\partial Y}{\partial t}$ = $(\frac{w \times v}{(1 + (uv)^{2}})$(0) + $(\frac{w \times u}{(1 + (uv)^{2}})$ (1) + (tan

$\frac{\partial Y}{\partial t}$ = $\frac{w \times u}{1 + uv^{2}}$ + tan

Evaluate u, v, and w at r = 0, s = 1, t = 2:

u = 0 + 1 = 1

v = 1 + 2 = 3

w = 2 + 0 = 2

Evaluate the partials at r = 0, s = 1, t = 2:

$\frac{\partial Y}{\partial r}$ = $\frac{2 \times 3)}{(1 + (1 \times 3)^{2})}$ + tan

$\frac{\partial Y}{\partial r}$ = $\frac{3}{5}$ + tan

$\frac{\partial Y}{\partial s}$ = $\frac{2(3 + 1)}{(1 + (1 \times 3)^{2})}$

$\frac{\partial Y}{\partial s}$ = $\frac{4}{5}$

$\frac{\partial Y}{\partial t}$ = $\frac{2 \times 1)}{(1 + (1 \times 3)^{2})}$ + tan

$\frac{\partial Y}{\partial t}$ = $\frac{1}{5}$ + tan

If u

The equation of the tangent plane to the graph of the function f of two variables at the point a, b, f(a,b)) is,

z = f(a, b) + f

Hence the function of the form,

f( x, y) = f(a, b) + f

Similarly, for a function of 3 variables, x , y, z,

f( x, y, z) = f(a, b) + f

Let f(x) = 0, be the given equation and x0 be an approximate root of the equation. If h is a small correction applied to the root, then x

The first approximation to the root x

(i.e) x

The second approximation is obtained by replacing x

(i.e) x

In general, x