Applications of Partial Differentiation and Successive Differentiation.
- What is a Jacobian of two or three variables and how do I calculate it? In one line The Jacobian J=∂(u,v)/∂(x,y) is the determinant of all first partial derivatives of u,v with respect to x,y ; it measures how a... 1
- What is Lagrange's method of undetermined multipliers for finding maxima and minima with a constraint? In one line To optimise f(x,y,z) subject to φ(x,y,z)=0 , form F=f+λφ and solve F_x=F_y=F_z=0 together with φ=0 . Explained simply Suppose yo... 1
- What are maxima and minima of a function of two variables and how do I find stationary points? In one line A function f(x,y) has a maximum (minimum) at a point if its value there is larger (smaller) than at all nearby points; such poin... 1
- How do I use the second derivative test with r, s, t to classify maxima, minima and saddle points? In one line At a stationary point compute r=f_xx , s=f_xy , t=f_yy : if rt-s²>0 it is a maximum when r<0 and a minimum when r>0 ; if rt-s²<0... 1
- What is a saddle point of a function of two variables and how do I identify it? In one line A saddle point is a stationary point that is a maximum along one direction and a minimum along another, so it is neither a maxim... 1
- How do I solve a word problem on maxima and minima, like the box with least surface area? In one line Write the quantity to optimise as a function, use the given condition to reduce it to two variables, then find and classify the... 1
- What are the nth derivatives of sin(ax+b) and cos(ax+b), and why does nπ/2 appear? In one line Dⁿsin(ax+b)=aⁿsin(ax+b+dfracnπ2) and Dⁿcos(ax+b)=aⁿcos(ax+b+dfracnπ2) : each derivative multiplies by a and shifts the angle by... 1
- What is successive differentiation and what are the nth derivatives of e^(ax), a^(mx) and (ax+b)^m? In one line Successive differentiation means differentiating a function again and again; for standard functions we spot the pattern and writ... 1
- How do I find the nth derivative of e^(ax) sin(bx+c) and e^(ax) cos(bx+c)? In one line Dⁿ[e^axsin(bx+c)]=rⁿe^axsin(bx+c+nφ) with r=√(a²+b²) , φ=tan⁻¹b/a ; each derivative multiplies by r and adds the angle φ . Expla... 1
- What are the nth derivatives of 1/(ax+b) and log(ax+b), and how are they derived? In one line Dⁿdfrac1ax+b=dfrac(-1)ⁿn! aⁿ(ax+b)ⁿ⁺¹ and Dⁿlog(ax+b)=dfrac(-1)ⁿ⁻¹(n-1)! aⁿ(ax+b)ⁿ ; signs alternate and factorials build up wit... 1
- How do I find the nth derivative of a rational function using partial fractions? In one line Break the rational function into simple partial fractions like A/x-a , then apply Dⁿdfrac1ax+b=dfrac(-1)ⁿn!aⁿ(ax+b)ⁿ⁺¹ to each p... 1
- What is Leibnitz's theorem for the nth derivative of a product and how do I apply it? In one line Leibnitz's theorem gives the n th derivative of a product: (uv)_n=Σ_(r=0)ⁿⁿC_r u_(n-r)v_r , which looks exactly like the binomia... 1
- How do I use Leibnitz's theorem to prove relations like (1−x²)y_(n+2) − (2n+1)x y_(n+1) − (n²+m²)y_n = 0? In one line First get a simple differential equation connecting y,y₁,y₂ , then differentiate it n times using Leibnitz's theorem on each pro... 1
- Show that the minimum value of \(u=xy+a^3\left(\frac1x+\frac1y\right)\) is \(3a^2\). u_x=y-a³/x², u_y=x-a³/y² . Stationary points: x²y=a³,\ xy²=a³ . Dividing, x=y ; then x³=a³⇒ x=y=a . Point (a,a) . r=u_xx=2a³/x³,\ s=u_xy=1,\... 1
- If \(y=a\cos(\log x)+b\sin(\log x)\), prove that \(x^2y_{n+2}+(2n+1)xy_{n+1}+(n^2+1)y_n=0\). y₁=-a/xsin(log x)+b/xcos(log x)⇒ xy₁=-asin(log x)+bcos(log x) . Differentiate: xy₂+y₁=-1/x[acos(log x)+bsin(log x)]=-y/x⇒ x²y₂+xy₁+y=0 . Lei... 1
- Find \(n^{th}\) derivative of \(\frac{x^2}{(x-1)(x-2)(x-3)}\). [The numerator is faint in the source scan; it appears to be \(x^2\).] (Solved taking the numerator as x² , as it appears in the scan.) Partial fractions: x²/(x-1)(x-2)(x-3)=frac Ax-1+frac Bx-2+frac Cx-3 . A=1/(... 1
- Examine the function \(f(x,y)=y^2+4xy+3x^2+x^3\) for extreme values. f_x=4y+6x+3x², f_y=2y+4x . Stationary points: f_y=0⇒ y=-2x . Then f_x=-8x+6x+3x²=3x²-2x=0⇒ x=0 or x=2/3 . Points: (0,0) and (2/3,-4/3) . r=f... 1
- Find \(n^{th}\) derivative of \(y=\sin2x\cos6x\). Using 2sin Acos B=sin(A+B)+sin(A-B) : y=1/2[sin8x+sin(-4x)]=1/2[sin8x-sin4x] . Using Dⁿsin(ax)=aⁿsin(ax+fracnπ2) : y_n=1/2[8ⁿsin(8x+fracnπ2)... 1
- Discuss the maxima and minima of \(x^3y^2(1-x-y)\). f=x³y²-x⁴y²-x³y³ . f_x=3x²y²-4x³y²-3x²y³=x²y²(3-4x-3y) ; f_y=2x³y-2x⁴y-3x³y²=x³y(2-2x-3y) . Stationary points (x, y ≠ 0): 4x+3y=3,\ 2x+3y=2⇒... 1
- If \(y=e^{m\sin^{-1}x}\), prove that \((1-x^2)y_{n+2}-(2n+1)xy_{n+1}-(n^2+m^2)y_n=0\) y₁=e^msin⁻¹x·frac m√(1-x²)⇒√(1-x²) y₁=my⇒(1-x²)y₁²=m²y² . Differentiate: (1-x²)2y₁y₂-2xy₁²=2m²yy₁ ; divide by 2y₁ : (1-x²)y₂-xy₁-m²y=0 . Lei... 1
- Find \(n^{th}\) derivative of \(y=e^{ax}\sin^2x\). y=e^ax·1-cos2x/2=1/2e^ax-1/2e^axcos2x . Standard results: Dⁿ(e^ax)=aⁿe^ax ; Dⁿ[e^axcos(bx+c)]=rⁿe^axcos(bx+c+nφ) where r=√(a²+b²),\ φ=tan⁻¹b... 1
- Discuss the maxima and minima of \((x^2+y^2+8x+6y+6)\). f=x²+y²+8x+6y+6 . Stationary points: f_x=2x+8=0⇒ x=-4 ; f_y=2y+6=0⇒ y=-3 . Stationary point (-4,-3) . Second derivatives: r=f_xx=2,\ s=f_xy=... 1
- If \(y=e^{\tan^{-1}x}\) then prove that \((1+x^2)y_{n+2}+[2(n+1)x-1]y_{n+1}+n(n+1)y_n=0\). y₁=e^tan⁻¹x·frac11+x²=frac y1+x²⇒(1+x²)y₁=y . Differentiate again: (1+x²)y₂+2xy₁=y₁⇒(1+x²)y₂+(2x-1)y₁=0 . Leibnitz's theorem (differentiate... 1
- Find \(n^{th}\) derivative of \(y=\frac{x}{1+3x+2x^2}\). 1+3x+2x²=(1+x)(1+2x) . Partial fractions: x/(1+x)(1+2x)=frac A1+x+frac B1+2x⇒ x=A(1+2x)+B(1+x) . x=-1:\ -1=-A⇒ A=1 ; x=-1/2:\ -1/2=B/2⇒ B=-1... 1