Chapitre 3 Dérivabilité des fonctions réelles
(2) On définit de même la dérivée `a droite que l'on note fd(x0). on applique le théor`eme pour la fonction log sur l'intervalle [x
MHT chap
6.2 Properties of Logarithms
Once we get the x2 by itself inside the log we may apply the Power Rule with u = x and w = 2 and simplify. log0.1(10x2) = log0.1(10) + log0.1(x2). Product Rule.
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4-Partial Derivatives and their Applications.pdf
them one is called partial derivative of z(x y) with respect to x denoted by one of the symbols *(i) u = (tan–1a) [log(x2 + y2)] + btan–1(y/x)
Partial Derivatives and their Applications
Appendix: algebra and calculus basics
28 sept. 2005 2. Logarithms convert products to sums: log(ab) = log(a) + log(b). ... The derivative of the logarithm d(log x)/dx
algnotes
CONTINUITY AND DIFFERENTIABILITY
(ii) The function y = f (x) is said to be differentiable in the closed interval [a b] The derivative of logx. w.r.t.
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DIFFERENTIAL EQUATIONS
An equation involving derivative (derivatives) of the dependent variable with log x. Example 18 A solution of the differential equation. 2.
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formulaire.pdf
Logarithme et Exponentielle : eln x = ln(ex) = x e−x = 1/ex. √ex = ex/2. (ex) y. = exy lim x→−∞ ex = 0 lim x→+∞ ... R`egles de dérivation.
formulaire
1. If log x2 – y2 = a then dy / dx = x2 + y2
1. If log x2 – y2. = a then x2 + y2. Solution : Take y /x = k → y = k x. → dy/dx = k. → dy/dx = y / x The derivative of an even function is always.
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%20Samples.pdf
New sharp bounds for the logarithmic function
5 mar. 2019 log(x + 1) i. e. log(1 + x) ⩾ 2x/(2 + x) for x ⩾ 0. ... and by another differentiation
9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. (ii) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation. (iii) Order of a differential equation is the order of the highest order derivative occurring in the differential equation. (iv) Degree of a differential equation is defined if it is a polynomial equation in its derivatives. (v) Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it. (vi) A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution. (vii) To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants. (viii) The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves. (ix) 'Variable separable method" is used to solve such an equation in which variables can be separated completely, i.e., terms containingxshould remain withdxand terms containingy should remain withdy.Chapter 9DIFFERENTIAL EQUATIONS
180 MATHEMATICS(x) A function F (x,y) is said to be a homogeneous function of degreen if
F (lx, ly)= ln F (x,y) for some non-zero constantl. (xi) A differential equation which can be expressed in the formdy dx= F (x,y) ordx dy = G (x,y), where F (x,y) and G (x,y) are homogeneous functions of degree zero, is called a homogeneous differential equation. (xii) To solve a homogeneous differential equation of the typedy dx = F (x,y), we make substitutiony =vx and to solve a homogeneous differential equation of the typedx dy = G (x,y), we make substitutionx =vy. (xiii) A differential equation of the formdy dx + Py = Q, where P and Q are constants or functions ofx only is known as a first order linear differential equation. Solution of such a differential equation is given byy (I.F.) =( )Q I.F.dx´ò+ C, whereI.F. (Integrating Factor) =Pdxeò.
(xiv) Another form of first order linear differential equation isdx dy + P1x = Q1, where P1 and Q1 are constants or functions ofy only. Solution of such a differential
equation is given byx (I.F.) =( )1Q × I.F.dyò+ C, where I.F. =1Pdyeò.9.2 Solved Examples
Short Answer (S.A.)
Example 1Find the differential equation of the family of curvesy = Ae2x + B.e-2x.Solutiony = Ae2x + B.e-2x
DIFFERENTIAL EQUATIONS 181dy
dx = 2Ae2x - 2 B.e-2x and2 2d y dx = 4Ae2x + 4Be-2x Thus2 2d y dx = 4yi.e.,2 2d y dx- 4y =0. Example 2Find the general solution of the differential equationdy dx=y x.Solutiondy
dx=y xÞdy y =dx xÞdy yò=dx xòÞ logy = logx + logcÞ y =cxExample 3Given thatdy
dx=yex andx = 0,y =e.Find the value ofy whenx = 1.Solutiondy
dx=yex Þ dy yò =xe dxò Þlogy =ex +c Substitutingx = 0 andy =e,we get loge= e0+ c, i.e., c = 0 (loge = 1)Therefore, logy= ex.
Now, substitutingx = 1 in the above, we get logy = eÞ y=ee.Example 4Solve the differential equationdy
dx +y x=x2. SolutionThe equation is of the type+P = Qdyydx, which is a linear differential equation.Now I.F. =1dxxò=elogx =x.
Therefore, solution of the given differential equation is182 MATHEMATICSy.x =2xx dxò, i.e. yx =4
4xc+Hencey =3
4x c x+. Example 5Find the differential equation of the family of lines through the origin. SolutionLety =mx be the family of lines through origin. Therefore,dy dx=mEliminatingm, we gety =dy
dx.x or xdy dx -y = 0. Example 6Find the differential equation of all non-horizontal lines in a plane. SolutionThe general equation of all non-horizontal lines in a plane is ax +by =c, wherea¹0.Therefore,dxa bdy+= 0.
Again, differentiating both sides w.r.t.y, we get22d xady = 0Þ2
2d x dy= 0. Example 7Find the equation of a curve whose tangent at any point on it, different from origin, has slopeyyx+. logy =x + logx +cÞlogyDIFFERENTIAL EQUATIONS 183Þ y
x=ex +c= ex.ecÞy x=k . ex y=kx . ex.Long Answer (L.A.)
Example 8Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x,y) of the curve is equal to the distance of P from thex - axis. SolutionLet the equation of normal at P(x,y) be Y -y =( )-X -dxxdy,i.e.,Y +Xdx
Therefore, the length of perpendicular from origin to (1) is2 1dx y xdy dx dy+ +ç ÷è ø...(2) Also distance between P andx-axis is |y|. Thus, we get2 1dx y xdy dx dy+ +ç ÷è ø= |y|21dxydyé ù
dy=ordx9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. (ii) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation. (iii) Order of a differential equation is the order of the highest order derivative occurring in the differential equation. (iv) Degree of a differential equation is defined if it is a polynomial equation in its derivatives. (v) Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it. (vi) A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution. (vii) To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants. (viii) The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves. (ix) 'Variable separable method" is used to solve such an equation in which variables can be separated completely, i.e., terms containingxshould remain withdxand terms containingy should remain withdy.Chapter 9DIFFERENTIAL EQUATIONS
180 MATHEMATICS(x) A function F (x,y) is said to be a homogeneous function of degreen if
F (lx, ly)= ln F (x,y) for some non-zero constantl. (xi) A differential equation which can be expressed in the formdy dx= F (x,y) ordx dy = G (x,y), where F (x,y) and G (x,y) are homogeneous functions of degree zero, is called a homogeneous differential equation. (xii) To solve a homogeneous differential equation of the typedy dx = F (x,y), we make substitutiony =vx and to solve a homogeneous differential equation of the typedx dy = G (x,y), we make substitutionx =vy. (xiii) A differential equation of the formdy dx + Py = Q, where P and Q are constants or functions ofx only is known as a first order linear differential equation. Solution of such a differential equation is given byy (I.F.) =( )Q I.F.dx´ò+ C, whereI.F. (Integrating Factor) =Pdxeò.
(xiv) Another form of first order linear differential equation isdx dy + P1x = Q1, where P1 and Q1 are constants or functions ofy only. Solution of such a differential
equation is given byx (I.F.) =( )1Q × I.F.dyò+ C, where I.F. =1Pdyeò.9.2 Solved Examples
Short Answer (S.A.)
Example 1Find the differential equation of the family of curvesy = Ae2x + B.e-2x.Solutiony = Ae2x + B.e-2x
DIFFERENTIAL EQUATIONS 181dy
dx = 2Ae2x - 2 B.e-2x and2 2d y dx = 4Ae2x + 4Be-2x Thus2 2d y dx = 4yi.e.,2 2d y dx- 4y =0. Example 2Find the general solution of the differential equationdy dx=y x.Solutiondy
dx=y xÞdy y =dx xÞdy yò=dx xòÞ logy = logx + logcÞ y =cxExample 3Given thatdy
dx=yex andx = 0,y =e.Find the value ofy whenx = 1.Solutiondy
dx=yex Þ dy yò =xe dxò Þlogy =ex +c Substitutingx = 0 andy =e,we get loge= e0+ c, i.e., c = 0 (loge = 1)Therefore, logy= ex.
Now, substitutingx = 1 in the above, we get logy = eÞ y=ee.Example 4Solve the differential equationdy
dx +y x=x2. SolutionThe equation is of the type+P = Qdyydx, which is a linear differential equation.Now I.F. =1dxxò=elogx =x.
Therefore, solution of the given differential equation is182 MATHEMATICSy.x =2xx dxò, i.e. yx =4
4xc+Hencey =3
4x c x+. Example 5Find the differential equation of the family of lines through the origin. SolutionLety =mx be the family of lines through origin. Therefore,dy dx=mEliminatingm, we gety =dy
dx.x or xdy dx -y = 0. Example 6Find the differential equation of all non-horizontal lines in a plane. SolutionThe general equation of all non-horizontal lines in a plane is ax +by =c, wherea¹0.Therefore,dxa bdy+= 0.
Again, differentiating both sides w.r.t.y, we get22d xady = 0Þ2
2d x dy= 0. Example 7Find the equation of a curve whose tangent at any point on it, different from origin, has slopeyyx+. logy =x + logx +cÞlogyDIFFERENTIAL EQUATIONS 183Þ y
x=ex +c= ex.ecÞy x=k . ex y=kx . ex.Long Answer (L.A.)
Example 8Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x,y) of the curve is equal to the distance of P from thex - axis. SolutionLet the equation of normal at P(x,y) be Y -y =( )-X -dxxdy,i.e.,Y +Xdx
Therefore, the length of perpendicular from origin to (1) is2 1dx y xdy dx dy+ +ç ÷è ø...(2) Also distance between P andx-axis is |y|. Thus, we get2 1dx y xdy dx dy+ +ç ÷è ø= |y|21dxydyé ù
dy=ordx- log(1+x^2) derivative
- log x base 2 derivative
- log(x^2+y^2) derivative
- log x^2 differentiation
- log(sec x^2) derivative
- log(sec x^2) derivative by first principle
- log tan x/2 differentiation
- log base x 2 differentiation