matesha
.doc-
VfubSet of numbers:
-
Definition of function:
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Intersection is
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Union is A∪B
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Disjoint is A∩B=ø
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Sequence is bounded below if
-
Sequence
is strictly decreasing if
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Sequence
is divergent if sequence
is oscillating or tending to infinity -
Bounded sequence
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Definition of limit of sequence is
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Sequence is monotone if
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Monotone convergence principle If
is an increasing sequence and bounded above, then it is convergent
sequence -
Theorem of Bolzano-Weiertrasse: every bounded sequence of complex numbers has A convergent subsequence
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Caushy sequence is Convergent, i.e. if given any ε>0, there exist N=N(ε)∈ℝ, depending on ε, such that
whenever m>n≥N -
Composite function is
-
Inverse function for
is 
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Limit of function in since “ε-δ” is
-
is
infinitesimal value -
is infinitely
large value -
Two Remarkable limits
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Function f(x) is continuous in point a
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Point of discontinuity of first kind (common definition) There are the finite limits exist:
,
,
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Points of discontinuity of second kind: If at least one limit
does not exist or equals to ∞ -
Points of discontinuity of first kind – jump point:
-
Points of removable discontinuity
third condition is not satisfied
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BOLZANO-WEIERSTRASS THEOREM 1 Suppose function y=f(x) continued on interval [a,b], so function y=f(x) is bounded on interval [a,b]
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BOLZANO-WEIERSTRASS THEOREM 2 Suppose function y=f(x) continued on interval [a,b], so function y=f(x) reaches maximum and minimum on interval [a,b]
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BOLZANO-CAUCHY THEOREM Suppose function y=f(x) continued on interval [a,b] and f(a)<0 and f(b)>0 or f(a)>0 and f(b)<0, so there exist such ξ∊(a,b): f(ξ)=0
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Derivative of function f(x) in point x=a is
-
,
where v is speed, s is distance, t is time Physical
interpretation of derivative -
Differentiation rules
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Differentiation rules
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Differentiation rules
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Differentiation rules
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If for
the inverse function exists and 
,
then -
Differentiation rules
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Differentiation rules
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Differentiation rules
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Chain rule of differentiation
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Derivative of n order Differential is main part of increment linearly with regard to 𝛥x: dy=f’(x)𝛥x
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Form of differential dy (in contradistinction to derivative) Is invariable
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L’hopital’s rule
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A function f(x) is said to have stationary point at x=a if
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A function f(x) is said to have a local maximum at x=a if
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A function f(x) is said to have a local minimum at x=a if
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If f’(x)>0 for every x<a in I and f’(x)<0 for every x>a in I, then
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If f’(x)<0 for every x<a in I and f’(x)>0 for every x>a in I, then
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Function f(x) is differentiable at every x∈I, and that f’(x)=0 and if f’’(x)<0, then
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Function f(x) is differentiable at every x∈I, and that f’(x)=0
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Function y=f(x) is concave upwards in interval [A,B], if for any x₁, x₂∈[A,b}
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Function y=f(x) is concave down in interval [A,B], if for any x₁, x₂∈[A,b}
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Inflection point is point of continued function dividing intervals of concavity (upwards and down)
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Prerequisite of inflection point
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Sufficient condition of inflection point: Recall (x₁, x₂, …, x𝓃) is critical point of function z and partial derivatives of second order exist and continued.
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Asymptote of y=f(x) is
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Let function y=f(x) is defined in ε of x₀ and lim f(x)=∞ as x→x₀-0 or lim f(x)=∞ as x→x₀+0
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Antiderivative is function F(x) such that F’(x)=f(x)
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Standard integrals
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Standard integrals
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Standard integrals
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Standard integrals
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Standard integrals
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Integration by substitution: If we maks a substitution x=g(u), then dx=g’du
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Integration by substitution
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Integration by parts
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Integration of irrationalities of kind
can be rationalize by substitution
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Integration of irrationalities of kind
can be rationalize by substitution:
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Integration of trigonometric functions
can be rationalized by substitution
-
Integration of trigonometric functions
can be rationalized by substitution -
Integration of trigonometric functions
can be rationalized by substitution -
Integration of trigonometric function
can be rationalized by substitution -
Non-integrable (in finite form) functions
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Non-integrable (in finite form) functions:
-
Non-integrable (in finite form) functions:
-
Non-integrable (in finite form) functions:
-
Non-integrable (in finite form) functions:
-
Non-integrable (in finite form) functions:
-
Non-integrable (in finite form) functions:
-
By the definite integral
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Newton-Leibniz Formulae: Suppose that a function F(x) satisfies F’(x)=f(x) for every x∈[A,B]. Then
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Property of definite integralLinearity
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Property of definite integral
Additivity:
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Property of definite integral
Monotonness:
If f(x)≤g(x),
and a<b, then 
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Property of definite integral
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Non-eigenvalue integral in half-interval [a,+∞)
-
If this limit exists and finite, then non-eigenvalue integral is convergent
If this limit does not exists or infinite, then non-eigenvalue integral is divergent
-
Particularly if non-eigenvalue integral
is divergent, but 
exists, so 
is Value
Principal of
Integral -
Multiplication of matrix
-
Transpose matrix
-
Row echelon form of matrix is
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Determinant of matrix 3x3 - Rule of triangles or Rule of Sarrus
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Determinant of matrix 2x2 :A=
-
det(A)=0 singular
-
det(A)≠0 non-singular
-
This matrix
Is augmented
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An nxn matrix A is said to be invertible if if there exists an nxn matrix B=
such that AB=BA=I -
Matrix I is Identity matrix if
-
Matrix with members
is is
Identity matrix -
,
then multiplicative inverse is
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Minor is
-
Cofactor
,
where 
-
where
K
is the adjoint of matrix A -
Trace of matrix is the sum of diagonal elements
-
Rank of a matrix is
-
Theorem of Kronecker-Capelli (Solution of system of linear equations)
System is inconsistent
System is consistent
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Cramer’s Rule (solution of system of linear equation using matrix analysis). Suppose that the matrix A is invertible. Then the unique solution of the system Ax=b, where A, x and b are variables of system of linear equations where
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Gaussian Elimination (Matrix Analysis)
The method of Gaussian Elimination reduces any set of linear equations to this triangular form by adding or subtracting suitable multiples of pairs of the equations.
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A vector is an object which has magnitude and direction
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For any vector u=(u₁, u₂) in ℝ², Norm of vector is real non-negative number
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Scalar products of vectors
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Economic interpretation of scalar products of vector
-
summary value of goods
where
- vector of volume of different goods,
- vector of prices of different goods
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Orthogonality condition for vectors: cos θ = 0 (by θ=π⁄2) ⇒ u₁v₁+u₂v₂+u₃v₃=0
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Colinearity condition for vectors v₁/u₁=v₂/u₂=v₃/u₃
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Orthogonal projection of vector. Suppose that u, a∈ℝ³. Then
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Scalar triple product of vector is
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Perpendicular distance D of a plane ax+by+cx+d=0 from a point (x₀, y₀, z₀) is given by
-
Suppose that
are vectors in a vector space V over R. By a linear combination of
the vectors 
,
we mean an
expression of the type 
,
where 
∈ℝ -
Suppose that
are vectors in a vector space V over ℝ:
are linearly dependent if
there exist 
,
not all zero, such that 
-
Suppose that
are vectors in a vector space V over ℝ:
are linearly independent if
the only solution of 
is given by 
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Suppose that
are vectors in vector space V or R. We say that {
is
a basis for V if the
followig two conditions are satisfied:
a){
=V
b)The
vectors 
are linearly independent.
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Dimension dim(V) of vector space V is maximum number of linearly independent vectors
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THEOREM of matrix of transformation . Matrixes A and A* of linear operator
over basis (
)
and basis ((
)
can be coupled as:
,
where C
is matrix of transformation from old basis to new basis.
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DEFINITION. “Quadratic form “
of n variables
or 
where every element is either squared variable, or scalar
multiplication of 2 different variables.
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Eigenvalue is solution of characteristics equation
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Quadratic form has canonical form, if
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Elements of Analytical Geometry: Relationship between Polar and Cartesian Co-ordinates: Superimpose one diagram upon the other a) X=r cosθ and y=r sin θ
b)
and 
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Elements of Analytical Geometry: A straight line is a set of points with cartesian co-ordinates (x,y) satisfying an equation of the form ax+by+c=0, where a, b and c are const.
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Elements of Analytical Geometry: Equation of a line passing through 2 given points
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Elements of Analytical Geometry: Distance between 2 points
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First order differential equation y’=f(x,y) is called incomplete if it does not contain (obviously) or the function itself with, or independent variable x: y’=f(x) or variable y: y’=f(y).
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The differential equation of the form y’=f(x)g(y) called a differential equation with multiple variables
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Linear differential equation of first order is an equation that is linear in the unknown function and its derivative, if
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First-Order Homogeneous Equations:
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Differential equation y’’+a₁y’+a₀y=f(x), where coefficients a₁, a₀ are const, linear differential equations of second order with fixed variables .
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Differential equation y’’+a₁y’+a₀y=f(x) where coefficients a₁, a₀ are constants, called linear differential equations of second order with fixed variables. If f(x)=0, then this equation called non homogeneous
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Differential equation y’’+a₁y’+a₀y=f(x) where coefficients a₁, a₀ are const, called called linear differential equations of second order with fixed variables If f(x)≠0, then this equation called homogeneous
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Fundamental system of solutions
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Functions y₁(x), y₂(x) called linear depended in interval (a,b) if there are exist constant numbers λ₁, λ₂ not equal to zero, such a λ₁y₁(x)+λ₂y₂(x)=0 for every x∈(a,b) . If this condition executes in a case when λ₁=0 and λ₂=0, so functions y₁(x), y₂(x) called linear independed in interval (a,b).
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Solution of homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=0: characteristic equation( to change in a homogeneous equation derivatives from initial function to K in appropriate power)
a₂=0
has 2 solution: k₁
and k₂,
then general solution is 
-
Solution of homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=0: characteristic equation( to change in a homogeneous equation derivatives from initial function to K in appropriate power)
a₂=0
has 1
solution: k₁=k₂=k,
then general solution is 
-
Solution of homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=0
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Solution of non-homogeneous linear differential equations of second order with fixed variables: y’’+a₁y’+a₂=f(x):solution of system of equations
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Caushy problem for differential equations: Caushy task – is a one of the main task in the theory of differential equations (ordinary and partial); it is consisted in
searching a solution (integral) of the differential equation, satisfying, so-called, initial conditions (initial data).
Let the function f(x) is a well defined value in D for . We should find a solution which satisfying initial conditions.
Caushy Task: If in D, function f (x, y) is continuous and has continuous partial derivatives , so for any pointin a neighborhood of corresponds only an unique solution
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Common member of series
:
Number
is sum of numerical series -
Sum of numerical series
is
common
member of numerical series -
Numerical series is convergent, if
exists and finite, then
-
Numerical series is divergent, if (lim is not exists or =∞)
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Property of convergence of numerical series
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Property of convergence of numerical series
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Property of convergence of numerical series:
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Prerequisite of convergence of numerical series
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Harmonic series is sum of infinite quantity of members reversed to serial natural numbers
-
Series is harmonic if because of each three members starting second
satisfies
to one rule
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Generalized harmonic series
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THEOREM OF D’ALAMBER (for series) If for series with positive members
is
valid, then the numerical series is convergent by l<1 and
divergent by l>1
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THEOREM OF CAUSHY (for series) Let for positive defined series
the limit exists
then if l<1, series is convergent, if l>1, series is
divergent, if l=1, convergence question is open. -
THEOREM OF LEIBNIZ (criteria of series convergence). Sign-alternative series
is convergent if
-
Sign-variable series
Its
members can be positive and negative -
THEOREM (sufficient condition of convergence for sign-variable series).
-
Series
is absolute
convergent
-
Series
is conditional
convergent
-
Consequence of Theorem of Leibniz Following Theorem of Leibniz we can evaluate error of sum calculation
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Determine the domain of the function:
=[1;+∞) -
Determine the domain of the function:
x≠±2
-
Determine the domain of the function:
=x>4
x≠7 -
Determine the domain of the function:
x>1
x≠3 -
Determine the domain of the function:
(0;2) -
Find the limit of the function:
=4 -
Find the limit of the function:
=10 -
Find the limit of the function:
=2/5 -
Find the limit of the function:
=-1/2 -
Find the limit of the function:
=3 -
Find the limit of the function:
=0 -
Find the limit of the function:
=5 -
Find the limit of the function:
=4 -
Find the limit of the function:
=8 -
Take derivative of the function:
=42xcos(7x^2-1) -
Take derivative of the function:
=5/2*√(x+3) -
Take derivative of the function:
=2xlnx+x -
Take derivative of the function:
=ctgx -
Take derivative of the function:
=-tgx -
Take second order derivative of the function:
=24x+10 -
Take second order derivative of the function: y = sinx=-sinx
-
Take derivative of the function: y = (sinx)3 =3(sinx)^2*cosx
-
Take derivative of the function: y = (lnx)2 =2lnx/x
-
Take derivative of the function: y =
=1/2√x -
Take derivative of the function: y =
=24cos(6x-1) -
Take derivative of the function: y =
=3*23x+2*ln2 -
Take derivative of the function: y =
=2*72x+5
*ln7 -
Take derivative of the function: y =
=3(5x^2
-4x +1)^2*(10x-4) -
Find the interval where the function is increasing:
=(-∞;-3)
(1;+∞) -
Find the interval where the function is decreasing:
(1;3) -
Find the extreme points of the function :
=xmax=
3 xmin=1 o y(1)-max and y(3)
-min -
Find the extreme points of the function :
xmax=4
xmin=2 y(2)-max and y(4) -min
-
Find the extreme points of the function :
x=3
y=3 -
Find the interval where the function is concave up:
(-1;+∞) -
Find the interval where the function is concave down:
(-∞;-1) -
Find the points of inflection of the function:
=-1 -
Find the interval where the function is concave up:
=(2;+∞) -
Find the interval where the function is concave down:
(-∞;2) -
Find the points of inflection of the function:
=2
-
Find the equations of the vertical asymptotes of the function:
-
x= -3 and x=3
-
Find the equations of the vertical asymptotes of the function:
=x=
-5 and x=5 -
Find the equations of the horizontal asymptotes of the function:
=Y=2 -
Find the equations of the horizontal asymptotes of the function:
=Y=3 -
Find the partial derivative
of
the function of two variables:
-
Find the partial derivative
of
the function of two variables:
-
Find the partial derivative
of
the function of two variables:
-
Find the partial derivative
of
the function of two variables:
-
Find the partial derivative
of
the function of two variables:
-
Find the partial derivative
of
the function of two variables:
-
Find the partial derivative
of
the function of two variables:
