BASIC DIFFERENTIATION
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A)
1) x⁵. 5x⁴
2) 5x². 10x
3) x¹⁶. 16x¹⁵
4) 3x⁷. 21x⁶
5) 1/x² +2√x. -2/x³ - 1/√x
6) x³ + 2x. 3x²+2
7) x⁴ + 6. 4x³
8) 3x² + 2x -5. 6x + 2
9) x³ -1. 3x²
10) 1/x. -1/x²
11) 1/x³. -3/x⁴
12) 1/x⁷. -7/x⁸
13) x + 1/x at x= 1. 0
14) 3x⁵ +7x⁴ - 2x²- x+6. 15x⁴+28x³-4x-1
15) (x² -3)³. 6x⁵- 36x³+54x
16) 2x⁴ - 6/x²+ 3x/³√x +2. 8x³+12/x³+2/³√x
17) 1/√(5x)+ √(5x)
18) (x² - 2)². 4x(x²-2)
19) (2x² +3x -4)/√x. 3√x+3/2√x +2/x√x
20) (x² - 2x)(x +1). 3x²-2x-2
21) 2 + 3/x +4/x² + 5/x³ -3/x² -8/x³ -15/x⁴
21) √x at x= 9. 1/6
22) x√x at x= 16. 6
23) ³√x at x= 27. 1/27
24) ⁷√x at x= 1. 1/7
25) 1/√x at x= 4. -1/16
26) 1/⁴√x at x= 1. -1/4
27) 3x³ - 2x +1. 9x² - 2
28) x√x + 1/√x + 3x. 3√x/2-1/(2√x³)+3
29) (2x²+3)/√x. {3(2x²-1)}/(2x√x)
30) (ax)ᵐ + bᵐ.
31) x³ +4x²+7x+2. 3x²+8x +7
32) 7x⁶+8x⁵-3x⁴+11x²+6x+7. 42x⁵+40x⁴- 12x³ + 22x + 6
33) 3+4x-7x²-√(2)x³+πx⁴-2x⁵/5+4/3 4 - 14x - 3√2 x² + 4πx³ - 2x⁴
34) 3/x⁵. -15/x⁴
35) (√x +1/√x)² and x≠0
36) √(x) - 1/√(x)
37) 1/x + 3/x² + 2/x³
38) (3x⁷+x⁵-2x⁴+x-3)/x⁴
39) 1 + x+ x²/2! + x³/3! +x⁴/4!
B)
1) e⁻ˣ - e⁻ˣ
2) e³ˣ 3e³ˣ
3) eᵃˣ⁺ᵇ. aeᵃˣ⁺ᵇ.
4) e³ˣ⁺¹. 3e³ˣ⁺¹
5) ₑx². 2xₑx²
6) ₑ√x. ( ₑ√x)/2√x
7) ₑ√(ax + b). a(ₑ√ax + b)/2√(ax+b)
8) e⁵ˣ⁺². 5e⁵ˣ⁺²
9) (e²ˣ + 1). 2e²ˣ
10) (e³ˣ+1)². 6e⁶ˣ+6e³ˣ
11) (e⁷ˣ+e²ˣ+3)/eˣ. 6e⁶ˣ+eˣ-3/eˣ
12) (e⁴ˣ-eˣ)/(e³ˣ-1). eˣ
C)
1) Log 2x. 1/x.
2) log x². 2/x
3) (1+ log5x) 1/x
4) log (3x+1). 3/(3x+1)
5) (logx)². 1/2(logx)
6) 5 log 9x. 5/9
D)
1) 3⁴ˣ. 3⁴ˣ log 3
2) 7ˣ + 5. 7ˣ log 7
3) a³ˣ +2x + 3 3a³ˣlog a + 2
4) (5ˣ+ 7³ˣ-1). 5ˣlog5+3.7³ˣlog 7
5) (2ˣ- 2³ˣ)/2³ˣ. -2log2/2²ˣ
6) (2ˣ + 3ˣ)²/6ˣ
E)
1) (2x+3)(x+1). 4x +5
2) (3x²+1)(x³+2x). 15x⁴+21x²+2
3) 2³ˣlogx. 2³ˣ(1/x +3 logx log 2)
4) 10ˣ x¹⁰. 10ˣx⁹(10+ x log 10)
5) eˣ(x + logx). eˣ(1+1/x +x + logx)
6) x³ eˣ x²eˣ(x+3x)
7) x⁵ logx x⁴(1+ 5 logx)
8) x⁵ eˣ + x⁶ logx x⁴eˣ(x+5)+ x⁵(1+ 6 logx
9) x⁵(3- 6/x⁹). 24/x⁵ + 15x⁴
10) 2ˣ x⁵ 2ˣx⁴(5+ x log 2)
11) 2x³ 2ˣ + 3eˣ log x 2ˣ.2x³ log 2+2ˣ.6x²+3eˣ(1/x + log x)
12) 2xᵃ - 3aˣ + c logx 2axᵃ⁻¹ - 3aˣ log a + c/x
F)
1) (2x+1)(3x²-1)(x³+2) 36x⁵+ 3x⁴+ 4x³+ 27x² - 14x -5
2) x² eˣ logx xeˣ(2log x+ x logx +1)
3) √x logx 10ˣ. 10ˣ(1/2√x logx + √x/x + √x logx log 10)
4) x log(x. eˣ) 5ˣ. 5ˣ(log(x.eˣ) + x+1 + x log(x. eˣ) log 5)
G)
1) (px+q)/(ax+b). (pb-aq)/(ax+b)²
2) (x²-3x+4)/(x+3). (2x²+6x-13)/(x+3)²
3) (x²-3)²/(x+2). (3x⁴+8x³+18x²+24x-9)/(x+2)²
4) 1/√(ax+b) -(a/2)/√(ax+b)³
5) (ax²+ bx+c )/(ax+b). (a²x²+ 2abx-ac)/(ax+b)²
6)(x²- 2x)/(x+1). (x²+2x-2)/(x+1)²
7) (x⁴-3x²-4)/(x-2). (3x⁴-3x²- 8x³+4)/(x-2)²
8) (2x+5)/(3x - 2). -19/((3x-2)²
9) (x² - 3)/(x+4). (x²+8x+3)/(x+4)²
10) (x⁵ - x +2)/(x³+7).(2x⁷+35x⁴+2x³-6x²-7)/ (x³+7)²
11) x²/(x+1). 2x/(x+1)²
12) x/(x²+1). (1-2x²)/(x²+1)²
13) (x²+2x+5)/(x²+2x+4). -2(x+1)/(x²+2x+4)²
14) (x³+2x)(x²+4).(x⁴+10x²+8)/(x²+4)²
15) (2+5x)²/(x³ - 1). -(25x⁴+40x³+12x²+100x+20)/(x³-1)²
16) x/(a² + x²). (a²-x²)/(a²+x²)²
17) (ax²+bx+c)/(px² +qx +f). (aqx²- bpx²+2afx- 2cpx+ bf- qc)/(px² + qx +f)²
18) 1/(ax+bx+c). -(a+b)/ (ax+bx+c)²
H)
1) |x|. x/|x|
2) √{(a² - x²)/(a² + x²)}. -2a²x/[(a²+x²).√(a⁴ - x⁴)]
3) 3/(3x²+5)³. -54x/(3x³+5)⁴
4) x/(a² - x³). (a²+2x³)/(a²-x³)²
5) y={x +√(x²+a²)}ⁿ. ny/√(x²+a²)
6) ³√(1+x²)⁴. (8x/3).³√(1+x²)
7) Find dy/dx if y = 5/p, p=x²/(x-1). {5(2-x)}/x³
8) (7x³ - 5x²+1)⁴. 4x(7x³-5x³+1)³ (21x- 10)
9) √(1+x²). x/√(1+x²)
10) 1/f(x) - f'(x)/{f(x)}²
11) f(x). f'(x)
12) Log(√x). 1/2x
13) if f(x)=7x²/4 find f´(1/7). 1/2
14) (x+2)/(x-2) find f′(-2). 1/4
15) (x+2)/(x²-3) find dy/dx at x=0. -1/3
16) (x-1)/(2x²-3x+5) find dy/dx at x= 3. -10/49
17) (x²+3)/(x³+2x) find dy/dx at x=1 -4/3
18) (e⁴ˣ-eˣ)/(e³ˣ-1). eˣ
I) Find dy/dx
1) x²+ y²= a² -x/y
2) x⁴ + x²y² + y⁴= 0. -x(2x²+y²)/y(2y²+x²)
3) x²/a² + y²/b². -b²x/a²y
4) x√y + y√x=√a. y(2√x+√y)/x(2√y+√x)
5) x/(x-y)= log{k/(x-y). (2y-x)/x
6) x²+y²+ 2gx + 2fy+c= 0. -(x+g)/(y+f)
7) x²y + xy²= x²+ y². (2x-2xy-y²)/ (x²+2xy-2y)
8) x= y log(x y). y(x-y)/x(x+y)
J) find dy/dx if
1) x= 3t - t², y= t+1. 1/(3-2t)
2)x= 3at/(1+t³), y= 3at²/(1+t³). (2t-t⁴)/(1-2t³)
K) Find dy/dx of
1) log x w.r.t (x² +2). 1/2x²
2) |x| w.r.t x (x ≠ 0). x/|x|
3) |2x -3| w.r.t x (x ≠ 3/2). 2|2x-3|/ (2x-3)
4) x⁵ w.r.t x³. 5x²/3
L)
1) {√(x²+1)-x}/{√(x²+1)+x}. 2(2x - (2x² +1)/√(x²+1)}
2) {(√x +1)√x(x√x -1)}/(x√x + x +√x) x-1
3) y= [√log x+ √{log x+√(logx.....)}] 1/{x(2y-1)}
4) log(4x -7). 4/(4x-7)
5) log(ax² + bx +c) (2ax+b)/(ax² + bx +c)
6) log ⁵√(x²+a²). 2x/5(x²+a²)
7) log(a+bx)³. 3b/(a+bx)
8) ₑ4(x² - log x +1) ₑ4(x² - log x +1) 4(2x - 1/x)
9) ₃2(x³ -1) ₃2(x³ -1) 6x² log 3
10) √(3x² - 6x +1) 3(x-1)/√(3x² - 6x +1)
11) 1/³√(6x⁵ - 7x³+1) (7x²-10x⁴)/√(6x⁵ - 7x³ +1)⁴⁾³
12) log⁷√{x+√(x²+a²)}. 1/7√(x²+a²)
13) {x - √(x² + a²)}ⁿ. n{x - √(x² + a²)}ⁿ/√(x² + a²)
14) log {log(log x)}. 1/x{log x log(log x)}
15) log {(x²+x+1)/(x² - x+1)}. 2(1-x²)/{(x²+x+1)(x²-x+1)}
16) 3/(2x²+5)². -24x/(2x²+5)³
17) x/√(a²- x²)⁵. (a²+4x²)/√(a²-x²)⁷
18) {√(x+1)+√(x-1)}/{√(x+1)-√(x-1)} 1+ x/√(x² -1)
19) x/{a- √(a²- x²)}. -{a(a+√(a²- x²))}/{x²√(a² - x²)}
20) {√(x²+a²)- √(x²- a²)}/{√(x²+a²)+√(x²- a²)}
21) √(ax² + bx+c). (2ax+b)/2√(ax² + bx +c)
22) (x²+4)²(2x³ -1)³. 2x(x²+4)(2x³-1)²(13x²+36x-2)
23) x²/√(4-x²). (8x-x³)/√(4-x²)³
24) (x-1)√(x² - 2x+2). (2x²-4x+3)/√(x²- 2x +2)
25) {(x³-1)/(2x³+1)}⁴. {36x²(x³-1)³/ (2x³+1)⁵
26) √{(a+x)/(a-x)}³. {x(3a²-x²)}/ √(a² - x²)³
27) √{(1-x)/(2+x)}. -3/{2√(1-x)√(2+x)³}
28) √{1+√x}. 1/{4√x+x√x}
29) ³√(1+x²)⁴. (8x/3) ³√(1+x²)
30) y=(u-1)/(u+1), & u=√x. 1/{√x(1+√x)²
31) y= (u²-1)/(u²+1) and u= ³√{x² +2). 8x/{3u(u²+1)²}
Prove:
1) If y=x+1/x prove x²dy/dx - xy+2=0
2) If y=√x +1/√x prove 2xdy/dx +y=2√x
3) y=1/(a-z) show dz/dy =(z - a)²
4) y=1+x+x²/2! + x³/3! +... to ∞s showthat dy/dx = y
5) Given that y= (2x-1)² + (2x -1)³f find dy/dx and the point on the curvefor which dy/dx = 0
6) y = 5x/(1- x)²/₃ + cos²(2x+1) show that dy/dx =5/4 (1-x)⁻⁵/³(3-x) - 2 sin(4x+2)
7) If f(x) = √{(x-1)(x+1)} prove f′(x)= 1/{(x+1)√(x²-1)}
8) If (x-2)/(x+2) show that 2x dy/dx = 1 - y²
9) y=x/(x+a) show x dy/dx = y(1-y)
10) x√(1+y) + y√(1+x) =0 then show that dy/dx = - 1/(1+x)²
11) y=√{(1-x)/(1+x) then show x(1-x²)dy/dx+y=0
12) Given that y= (2x-1)² + (2x -1)³ find dy/dx and the point on the curve for which dy/dx = 0
13) find the cordinates of the point of the curve y=x/(1-x²) for which dy/dx=1
14) If y log x= x - y Prove that dy/dx=( log x)/(1+ log x)²
15) x √(1+y) + y√(1+x)= 0 prove (1+x)² dy/dx +1 = 0
16) xy - log y = 1 then prove y²+ (xy-1)dy/dx = 0
17) If y log x= x+y, prove dy/dx= (log x -2)/(log x -1)²
18) If √(1-x²)+ √(1-y²)= k(x -y) show dy/dx= √{(1-y²)/(1-x²)}
19) eˣʸ= 4(1+xy) show dy/dx= -y/x
20) If y= √[log x +√{logx+ √(logx+... undefined, prove that, dy/dx= 1/{x(2y-1)}
21) If y= {x + √(x² -1)}ᵐ , prove that (x² -1) (dy/dx)² = m²y²
22) If xᵐ yⁿ= (x+y)ᵐ⁺ⁿ prove that dy/dx= y/x
23) x⁹y⁸ = (x+y)¹⁷, show that dy/dx = y/x
24) If 2x = y¹⁾ᵐ + y ⁻¹⁾ᵐ prove that (x² -1) (dy/dx)² = m²y²
25) If x= 2t/(1+t²), y= (1-t²)/(1+t²), Prove that dy/dx + x/y = 0
26) If y= log √{2x+ √(4x²+a²)}, show that dy/dx= 1/√(4x² +a²)
27) If y= √[x+√{x+√(x+....)}], Show that dy/dx= 1/(2y-1)
28) If y= (x-2)/(x+2), then show that 2x dy/dx = 1 - y²
30) If y= x + x³/3 + x⁵/5 +....., Show that dy/dx= 1/(1-x²)
31) If y= 1+ x+ x²/2! + x³/3! + x⁴/4! + ..... xᵐ/n! , Prove dy/dx + xᵐ/n!= y
32) y=√(x/m) +√(m/x), show that 2xy dy/dx = x/m - m/x,
33) If f(x)= |x| + |x -1|, then find dy/ dx
34) If y= x⁵ show x dy/dx - 5y = 0
35) If f(x)= x|x|, show f'(x)= 2|x|
36) If f(x)= |x-2| +|x-4| show f'(3)=0
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2nd Order Differentiation
y₂ as d²y/dx² and y₁ as dy/dx
A) Find d²y/dx²:::
1) 4x³ - 3x²+1. 24x - 6
2) 1/(x+a). 2/(x+a)³
3) xˣ. xˣ{(1+ logx)² + 1/x}.
4) x³ + tanx. 6x + 2sec²x tanx
5) sin(logx). -[sin(logx)+ cos(logx)]/x².
6) log(sinx). - cosec²x.
7) eˣ sin 5x. 2eˣ(5 cos 5x - 12 sin 5x).
8) e⁶ˣ cos 3x. 9e⁶ˣ(3 cos 3x - 4 sin 3x).
9) x³ logx. x(5+ 6 logx)
10) x cosx. - x cosx - 2 sinx.
11) log (logx). -(1+logx)/(x logx)².
12) Log(1-x). -1/(1-x)²
13) Log{x²/e²}. -2/x²
14) If x= 4t²+ 5, y= 6t² + 7t +3. -7/64t³
B) Prove::
1) If y= √(3x+2). Then y y₂ + y₁²=0
2) If y= y= (x+ √(1+x²)ⁿ, Prove that (1+x²) y₂ + x y₁= m²y.
3) If x²/a² + y²/b²= 1, show y₂ = - b⁴/(a²y³).
4) If ax² + 2hxy + by²= 1, show y₁ = -(ax+hy)/(hx+by) and y₂ =(h²-ab)/(hx+ by)³.
5) If y(1-x)= x², show (1-x) y₂- 2 y₁= 2.
6) If y= √(x+1) + √(x-1), then show that (x² - 1)y₂+ x y₁= y/4.
7) If y= px + q/x², then show that x²y₂ + 2xy₁= 2y.
8) If y² = p(x) be a polinomial in x of degree ≥ 3, then obtain 2y³y₂ = y²p"(x) - 1/2 {p'(x)}².
9) y= xˣ, show y₂ - 1/y (y₁)² - y/x= 0.
10) If (x-a)²+ (y -b)²= c², show that [{1+(y₁)²}³/²]/y₂ is a constant independent of a and b.
11) If y= (ax+b)/(cx+d), show that
2 dy/dx. d³y/dx³ = 3(d²y/dx²)².
C) PROVE:
1) If y= A cos nx + B sin nx, then y₂ + n²y= 0.
2) If y= 3 cos x + 2 sin x, then y₂ + y= 0.
3) Find A and B so that y= A sin 3x + B cos 3x, then d²y/dx²+ 4 dy/dx + 3y = 10 cos 3x.
4) If y= cotx, show y₂ + 2yy₁= 0.
5) If y= x + tanx, show that
A) cos 2x y₂ -2y + 2x= 0.
B) cos²x y₂ -2y + 2x= 0.
6) If y= Cosecx + cotx, show sinx y₂ = y².
7) If y= a cos(logx) + b sin(logx), show x²y₂ + x y₁+ y= 0.
8) If y= 3 cos(logx) + 4 sin(logx), show x²y₂ + x y₁+ y= 0.
9) If y= tanx + secx, show (1- sinx)² y₂ = cosx.
10) y= tanx show y₂ = 2yy₁
11) If x = tan(1/a log y), show (1+ x²)y₂ + (2x - a)y₁ = 0.
12) If x = sin(1/a log y), show (1- x²)y₂ - xy₁ - a²y = 0.
13) If y= sin(sinx), show y₂ + tanx y₁ + y cos²x = 0.
14) If y= sin(logx), show x² d²y/dx² + x dy/dx + y= 0.
D) PROVE::
1) If y= x³ log(1/x), then xy₂- 2 y₁+ 3x²= 0.
2) If y= x³ logx , then d⁴y/dx⁴= 6/x.
3) If y= log(sinx), show d³y/dx³= 2 cosx cosec³x.
4) If y= log{x + √(x² + a²)}, show that (a² + x²) y₂ + xy₁= 0.
5) If y= {log{x + √(x² + 1)}², show that (1+ x²) y₂ + xy₁= 4
6) If y= log(ax+b), show y₂= - a²/(ax + b)².
7) If y= (logx)/x, show x³y₂ - 2logx +3= 0.
8) If y= log{√(x-2) + √(x+2)}, then show (x² -4)y₂+ x y₁= 0.
9) If y= log{x +√(x²+1)}, then show that (x² -1)y₂ + x y₁= 0.
10) If u= v³ log(1/v), show v d²u/dv² - 2 du/dv + 3v² = 0.
11) If y= x log{x/(a+bx)}, show x³y₂ = (x y₁ -y)².
12) If y= log(1+ cosx), show d³y/dx³ + d²y/dx². dy/dx = 0.
E) PROVE :::
1) y= aeᵐˣ+ beᵐˣ, show y₂ + = m²y.
2) y= xᵐ eⁿˣ, show y₂ ={m(m-1)xᵐ⁻² + 2mnxᵐ⁻¹ + n²xᵐ} eⁿˣ.
3) If eʸ(x+1) = 1, show y₂ = (y₁)².
4) If y= e⁻ˣ cosx, show y₂= 2e⁻ˣ sinx
5) If y= eˣ cosx, show y₂= 2eˣ cos(x +π/2).
6) If y= eˣ (ax+b) show y₂- 4y₁ + 4y= 0.
7) If y= e⁻ᵏᵗ cos(pt +c), show y₂ +2k y₁ + ny²= 0. Where n² = p² + k².
8) If y= a eˣ + be⁻ˣ show y₂ - y₁ -2y= 0.
9) If y= eˣ(sinx + cosx) show y₂ - 2y₁ + 2y= 0.
10) If y= ₑ a cos⁻¹x show (1-x²) d²y/dx² - x dy/dx - a²y = 0.
11) If y= 509 e⁷ˣ + 600e⁻⁷ˣ show y₂ = 49y
12) If y= 3e²ˣ + 2e³ˣ show y₂ - 5 y₁ +
6y= 0.
F) PROVE
1) If y= tan⁻¹x, show. (1+x²)y₂+2x y₁ = 0.
2) If y= (tan⁻¹x)², show. (1+x²)²y₂+ 2x(1+x²) y₁ = 2.
3) ₑ m cos⁻¹x, show. (1- x²)y₂ - x y₁ = m²y.
4) If y= (sin⁻¹x)², show. (1- x²)y₂ - x y₁ = 2.
5) If cos⁻¹(y/b)= n log (x/n) show. x²y₂ + x y₁ + n²y = 0.
6) If sin⁻¹x= y, show. y₂ = -x/√(1-x²)³
7) If y= sin⁻¹x show.(1-x²) y₂ - y₁=0
8) If log y= tan⁻¹x show.(1+ x²) y₂ +(2x -1) y₁=0.
G) PROVE:::
1) If x= at², y= 2at. 1/(2at³)
2) If x= a(1- cos t), y= a(t+ sint) at t= π/2. -1/a
3) If x= sint, y= sin(pt), show (1-x²)y₂ - xy₁ + p²y = 0.
4) If x= a(t + sint), y= a(1- cost), then show 4ay₂ = sec⁴(t/2)
5) If y= a(t + sint), x= a(1- cost), then show y₂ = -1/a at t=π/2
6) If x= a(1 + cost), y= a(t+ sint), then show y₂ = -1/a at t=π/2.
7) If x= a(t + sint), y= a(1 + cost), then show y₂ = - a/y².
8) If x= (1-t)/(1+t) and y= 2t/(1+t), show that y₂ = 0.
9) If x= t + 1/t and y= t - 1/t, then show at t=2 is y₂ = -32/27
10) x= a cost + b sint and y= a sint - b cost, show y² y₂ - xy₁ + y = 0.
11) x= a sin t - b cost and y= a cost + b sin t, show y₂ = -(x²+y²)/y³.
12) If x= a cos³t, y= a sin³t, show y₂ = 1/3a sec⁴ t cosec t.
13) If x= a sect and y= b tant show d²y/dx² = - b⁴/(a²y³).
14) If x= a(cost + t sint), y= a(sint - t cost), show d²y/dx²= sec³t/at.
15) If x= a cost, y= b sint , show that d²y/dx² = - b⁴/(a²y³).
16) If x= a(1- cos³t), y= a sin³t, show d²y/dx²= 32/27a at t= π/6.
17) If x= cost, y= sin³t, show yd²y/dx² +(dy/dx)²= 3 sin²t(5cos²t -1).
18) If y= 2 cost - cos 2t, y= 2 sint - sin 2t, at t=π/2 show y₂ = -coty cosec²y.
H)
MISCELLANEOUS QUESTIONS:
PROVE::
1) If 2x= y¹⁾ᵐ+ y⁻¹⁾ᵐ, show (x²-1)y₂ + x y₁= m²y.
2) If 2x= y¹⁾⁵ + y⁻¹⁾⁵, show (x²-1)y₂ + x y₁= 25y.
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