June 6, 20XX
Today, the world witnessed an unprecedented alliance between intergalactic forces as they united against a common threat. I, a mere observer from a different time period, found myself in the midst of a grand spectacle of heroism and warfare. The skies were filled with tension as the combined military might of Earth, the Mushroom Kingdom, and Mobius prepared for an aerial assault against the formidable doomship, a floating behemoth of destruction.
The call to arms echoed across the lands as Mario, Luigi, Sonic, and Yoshi, iconic heroes from their respective realms, were summoned to lead this daring operation. These extraordinary individuals were entrusted with the task of mounting an aerial assault on the doomship, striking a decisive blow against the forces of evil.
As I watched from a safe distance, the scene before me was a symphony of preparation. The air crackled with energy as aircraft of various designs and origins filled the sky, forming a united front against the impending menace. The intergalactic Earth forces, Mushroom Kingdom's aerial squadron, and Mobius' airborne battalion worked in unison, displaying an awe-inspiring display of coordination and unity.
Mario and his trusted brother Luigi stood at the forefront, their courage shining through their determined expressions. Equipped with specialized aircraft, they awaited the signal to launch their assault. Sonic, the lightning-fast hedgehog, exuded confidence as he readied himself for the impending battle. Yoshi, the trusty dinosaur, mounted his own unique flying contraption, ready to unleash his aerial prowess.
Meanwhile, Shadow, with his command in the G.U.N, took charge of coordinating the ground forces. He understood the importance of countering the blue spikey robot, an adversary that demanded a tactful approach. Shadow's deep sense of responsibility and his unwavering dedication to the cause were evident as he directed the troops, ensuring that every move was calculated and precise.
As the skies darkened with storm clouds of impending conflict, the first assault wave took flight. Allied air forces, a combination of Earth's most advanced military aircraft, Mushroom Kingdom's loyal airships, and Mobius' nimble flyers, clashed against the overwhelming might of King Bowser's and Dr. Eggman's air troops. Explosions illuminated the sky, forming a breathtaking display of fireworks of chaos and heroism.
The battle was a testament to the bravery and skill of these remarkable heroes and their allies. Mario and Luigi maneuvered their aircraft with unmatched finesse, raining down firepower upon the doomship's defenses. Sonic, a veritable blur, darted through the air, striking with lightning speed and precision. Yoshi, unleashing his unique abilities, proved to be a formidable aerial force, toppling enemy airships with his mighty kicks and fire breath.
The stakes were high, and the battle waged on relentlessly. The doomship unleashed a barrage of devastating weapons, testing the mettle of the allied forces. But their determination and unwavering spirit shone through, as they defied the odds and pressed forward, inch by hard-fought inch.
As the chaos raged above, Shadow coordinated the ground forces, ensuring their efforts were focused and strategic. His sharp intellect and unparalleled combat prowess were on full display as he rallied troops and devised strategies to counter the relentless advance of the blue spikey robot. The ground shook as battles unfolded, the clash between Shadow's forces and the robot creating seismic waves of power and destruction.
The air assault continued, a fierce dance of skill, courage, and resilience. Explosions rocked the sky, mingling with the roars of victory and the cries of defeat. The doomship, once an invincible fortress, now teetered on the brink of annihilation, its defenses weakened by the united onslaught of the heroes and their allied forces.
Math Notes: (filled with academic musings)
d/dx (c)=0
d/dx [cf(x)]=cf'(x)
d/dx [f(x) + g(x)]=f'(x)+g'(x)
d/dx [f(x)+g(x)]=f'(x)+g'(x)
d/dx [f(x)-g(x)]=f'(x)-g(x)
d/dx [f(x)-g(x)]=f'(x)-g'(x)
Product Rule: [f(x)g(x)]=f(x)g'(x)+g(x)f'(x)
Quotient Rule: d/dx[f(x)/g(x)]=g(x)f'(x)-f(x)g(x) / [g(x)]^2
Chain Rule: d/dx f(g(x))=f'(g(x))g'(x)
Power Rule: d/dx (x^n)=nx^n-1
...
d/dx (sinh^-1 x)= -1 square root 1+x^2
d/dx (cosh^-1 x)= -1 square root x^2 - 1
d/dx (tanh^-1 x)= -1/1-x^2
d/dx (csch^-1 x)= -1/1x1 square root x^2 + 1
d/dx (sech^-1 x)= -1/x square root 1-x
d/dx (coth^-1x)= 1/1-x^2
...
d/dx (e^1)= e^1
d/dx (a^1)= a^1 ln(a)
d/dx ln |x| = 1/x
d/dx (log(n)x)=1/xlna
...
d/dx (sin x)= cos x
d/dx (cos x)= -sin x
d/dx (tan x)= sec' x
d/dx (csc x)= -csc x cot x
d/dx (sec x)= sec x tan x
d/dx (cot x)= -csc^2 x
...
d/dx (sin^-1 x)= 1/square root 1-x^2
d/dx (cos^-1 x)= -1/square root 1-x^2
d/dx (tan^-1 x)= 1/1+x^2
d/dx (csc^-1 x)= -1/x square root x^2 - 1
d/dx (sec^-1 x)= 1/ x square root x^2 - 1
d/dx (cot^-1 x)= -1/1+x^2
...
d/dx (sin hx) = cos hx
d/dx (cos hx) = sin hx
d/dx (tan hx) = sec h^2 x
d/dx (csc hx)= -csc hx cot hx
d/dx (sec hx)= -sec hx tan hx
d/dx (cot hx) = -csch'x
...
1. dy/dx
y=-12x^7/4
dy/dx (-12x^7/4)
(7/4 X -12x^7/4-1)
-21x^3/4
2. f'(x) = (x^2) cos (x)
d/dx (x^2) cos (x)
f'(x) = 2x(-sin (x))
f(x) (d/dx g(x)) + g(x) (d/dx f(x))
f(x)=x^2
g(x)=cos(x)
f' = x^2 d/dx(cos(x))+(cos(x)) d/dx(x^2)
x^2 (-sin x) + cos(x) 2x
or x^2 (-sin x) + 2x cos(x)
3. y=(3t^4)/2t-5
g(t) dy/dt(f(t)) - f(t) dy/dt(2t-5) / (2t-5)^2
(2t-5)(12t^3) - (3t^4)(2) / (2t-5)^2
24t^4 - 60t^3 - 6t^4 / (2t-5)^2
=18t^4 - 60t^3 / (2t-5)^2
...
Derivatives
f(x)=x^-4 - 9x^-3 + 8x^-2 +12
f'(x)= -4x^-5 + 27x^-4 - 16x^-3
Derivatives
f(x)= square root x + 8 (3 square root) x + 10 (5 square root) x^3
f'(x)=1/2x^1/2-1 + 8/3^1/3-1 + 10(3/5)x^3/5-1
f'(x)=1/2x^-1/2 + 8/3(x)^-2/3 + 6x^-2/5
Derivatives
f(x)= square root x + 9 (3 square root) x + 11 (5 square root) x^4
f'(x)= 1/2(x)^1/2-1 + 3x^1/3-1 + 11(4/5)x^4/5-1
f'(x)= 1/2(x)^-1/2 + 3x^-2/3 + 44/5(x)^-1/5
Derivatives
f(y) = (y-4)(2y+y^2)^5
Chain Rule, Product Rule
f'(y)=(y-4) d/dx(2y+y^2)^5 + d/dx(y-4)(2y+y^2)^5
=(y-4)[5(2y+y^2)^4 (2+2y)] + (1)(2y+y^2)^5
Derivatives
(y-5)(3y+y^3)^6
f'(y)=(y-5) d/dx(3y+y^3)^6 + d/dx(y-5)(3y+y^3)^6
(y-5)[g(3y+y^3)^5 (3+3y^2)] + (1)(3y+y^3)^6
Derivatives
(y-6)(4y+y^4)^5
f'(x)=(y-5)[5(4y+y^4)(4+4y^3)] + (1)(4y+y^4)^5
f'(x)=(y-5)[5(4y+y^4)(4+4y^3)] + (1)(4y+y^4)^5
Derivatives
f(n)=4n^3 X tan(6n^2 -7n)
(4n^3) d/dx[tan(6n^2-7n)] + d/dx(4n^3)(6n^2-7n)
(4n^3)[sec^2(6n^2-7n)(12n-7)]+(12n^2)(tan(6n^2-7n)
Derivatives
f(n)=5n^4 [tan(7n^3-8n)]
(5n^4) d/dx[tan(7n^3-8n)] + d/dx(5n^4)[tan(7n^3-8n)]
(5n^4)[sec^2(7n^3-8n)(21n-8)]+(20n^3)[tan(7n^3-8n)]
Derivatives
f(t)= t^2/3 / t+t^3
(t+t^3) d/dx(t^2/3)-(t+t^3) d/dx(t+t^3) / (t+t^3)^2
(t+t^3) 2/3(t^-1/3)-(t+t^3)(1) / (t+t^3)^2
Derivatives
f(theta) = cos(2theta) / cos(17theta)
g(x)f'(x)-f(x)g'(x) / g(x)^2
cos(17theta) d/dx(cos(2theta))-(cos(2theta)) d/dx(cos(17theta)) / cos^2(17theta)
(cos(17theta)(-2sin(2theta))-(cos(2theta)(-17sin(17theta)) / cos(17theta)^2