Balloon Calculus Gallery




At present, you generally need to click on the pictures below to get the full context and explanations.




Differentiate a product

Differentiate a quotient

Differentiate a composite function

Implicit differentiation

Integration by parts

Integration by substitution / change of variable

Integration by internal substitution

Separation of variables

Integrating factor

Fundamental theorem of calculus








Differentiate a product

product rule

... is the product rule. The straight lines differentiate downwards (integrate up) with respect to x (or whatever the main, explicit variable). And then, because of the product rule, the whole of the bottom row is the derivative (with respect to x) of the whole of the top row. Choosing legs crossed or uncrossed is unimportant when differentiating but crucial for integration by parts.



So differentiation is a downwards journey, and if you want to expand the picture with equals signs (often a good idea) then it's probably also clockwise...
differentiate a product



Some of the following examples have the chain rule wrapped inside the product rule...


production level and price




-8x e^(-x^2)



sqrt(t) (1 - t^2)



sqrt(t) (1 - t^2)



total & average income



total & average income



differentiate x^2 e^(-3x)



differentiate x^2 (x - 2)^2



differentiate 3x e^(2x)



differentiate cos^4(3t) sin(3t)










Differentiate a quotient


differentiate a quotient



differentiate (2x - 3)^2 / (x^3 - 7)^3




differentiate (1 - x^2)/(x^2 + 1)^2



differentiate (x + 5)/(2x - 4)



differentiate tan x ln|2x + 5|



differentiate -4x/(x^2 - 1)^2



differentiate (1 + x^2) / (1 + 2x^2)



differentiate 5 cos(x) / (4 - sec(x))



differentiate x/(x + 2)^2



differentiate x^2/e^(-2x)



differentiate tan t (exponential form)



differentiate sec x



differentiate ln|x| / sqrt(x)



differentiate sqrt(7x - 2) / sqrt(5x + 3)



differentiate (4x + 3)^7 / (5x - 2)^7



differentiate 3x / (x^2 + 4)



differentiate (4 - x^2) / (x^2 + 4)^2



differentiate (1 - s^2) / (1 + s^2)^2








Differentiate a composite function

chain rule
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (which is the inner function of the composite expression which is subject to the chain rule).

As with the product rule, differentiation is generally a clockwise journey...
differentiate a composite



differentiate tan^2(x^3)



differentiate 5/(x^3) - 3/(2-7x) + 3x^5 - 15



differentiate sec^3(pi/2 - x)



differentiate e^(kx)



differentiate e^(xy) wrt x




differentiate sqrt(x + sqrt(x + sqrt(x)))
differentiate sqrt(x + sqrt(x + sqrt(x)))




differentiate arcsin(2x)^3



differentiate [ sin( 1 + { cos( 1 + [tan(1 + x)]^4 ) }^3 ) ]^2




differentiate [(4x + 3) / (5x - 2)]^7



differentiate e^(sec x)



differentiate (4 - 2x^2)^6



differentiate ln (sin^2 x)



differentiate ln (x^2 + 5)



differentiate (6x^4 - 6x^(-3) - 2x + 5)^(-2)



differentiate - ln |cos x|



differentiate ln |8x^3 - 7x + 2.78|



Related rates
shadow against a wall



revenue given rate for cost



ladder sliding down wall



rate y 12 times rate x




area of inkstain



track climbing aircraft



aeroplanes converging




piston, crank, connecting rod




rotating light-house beam



coffee filling in truncated cone



rotating spot-light



ladder towards wall



inflating balloon



conical sand pile



Double composite
d/dx (x^2)^(e^x)



d/dt (5 - 3t)^(4t)



d/dx x^(1/x)



d/dx [f(x)]^[g(x)]



d/dx x^(sin x)



d/dx 10 x^(ln x)



∂/∂x (x^2 - y^2) / (4xy^3)








Implicit differentiation
d/dt sqrt(x^2 + y^2)
differentiate x^2y + xy^2 = 6




differentiate 4x^3 + 2xy - y^3 = 1/2




differentiate xyz - 3yz^2 - 4xy^2 = 0




differentiate x e^y = y - 1



e^(x^2) + y^6 = 2x cos y



x^2 + y^2 = 25



differentiate x^3 y / (1 - xy)



differentiate 2x^2 y = sin(2x)



differentiate x^2 y = x + y^2




differentiate sin(3x) cos(2y) = 6



differentiate 3x^2 + 4xy^3 = 9



differentiate 6x + 10y dy/dx - 6 + 20 dy/dx = 0



differentiate x^2 - xy + (3/4)y^2 = 7



question: OCR GCE Maths 4724/01 q6
working: OCR GCE Maths 4724/01 q6









Integration by substitution / change of variable
integration by substitution



integrate 1 / (e^(2x) + 1)



integrate z^3 e^(-1/2 z^2)




integrate e^(2x) sqrt(7 + e^(2x))



integrate e^(kx)



integrate x e^(x^2)



integrate x^(24) sqrt(a^2 - x^(25)) from 0 to a^(2/25)



integrate (e^z + 21) / (e^z + 21z) from 0 to 1



integrate sqrt(1 + sqrt(x))

integrate sqrt(1 + sqrt(x))




integrate 1 / (9 sin(x)^2 + 4 cos(x)^2)



integrate 1 / [x (1 - x^(1/4))]



integrate x/sqrt(3 - x)



integrate tan x



integrate 2x / (3 + x^4)



integrate e^(sin x) sin(2x)



integrate x^2 / sqrt(2 + x^3)



integrate p(p + 1)^5



integrate t / (t^2 + 2)



integrate t3 sqrt(t^2 + 1)
integrate t3 sqrt(t^2 + 1)




integrate x^9 cos(x^5)



integrate x^9 cos(x^5)




integrate sqrt(x) / (1 + x)










Integration by internal substitution
integration by internal substitution



integrate 1/(x^2 + 25)




integrate 1/2 x^2 sqrt(1 + x^2)




integrate sqrt(1 + 1/x^2)



integrate sqrt(1 + 1/x^2)




integrate sqrt(1 + 4x^2)



integrate sqrt(1 + 4x^2)




integrate sqrt(1 + (x/a)^2)



integrate sqrt(1 + (x/a)^2)




integrate 1 / (c^2 + y^2)^(3/2)



integrate 1 / (c^2 + y^2)^(3/2)




integrate 1 / sqrt(9 - x^2)



integrate 1 / sqrt(9 + x^2)



integrate sqrt(4 - x^2)




integrate 1 / sqrt(49 + x^2)



integrate 1 / (x^2 + 1)^2



integrate 1 / (x^2 sqrt(x^2 - a^2))



integrate x^3 / sqrt(16 + x^2)



integrate x^3 / sqrt(16 + x^2)



integrate 1 / sqrt(1 - x^2) or differentiate arcsin



integrate 1 / sqrt(x^2 - 9/4)



integrate 1 / sqrt(x^2 - 9/4)



Weierstrass substitution
integrate 2 / (2 cos x + sin x)




integrate sec(x)




Weierstrass integral of cosec x









Differentiate an inverse trig function
differentiate and inverse trig function



differentiate arctan(4 - 5x)



differentiate arcsec(2 sqrt(t))




differentiate arcsec(2 sqrt(t))



differentiate arcsec(2 sqrt(t))








Integration by parts
integration by parts



sqrt(1 - x^2)




5x e^(-x)




x arctan x




integrate x^3 sqrt(x^2 + 1)




integrate (sin^2 x) / (cos^5 x)




integrate (1/3)u sin u



integrate 3 e^(2 sin(x)/3) sin(x)



integrate 3 e^(2 sin(x)/3) sin(x) - 2 e^(2 sin(x)/3) cos(x)^2



integrate sqrt(a^2 - x^2)



integrate (x + 2) (x + 3e^(-x/3))



integrate 5x e^(4x)



integrate x e(-x/4)



integrate sqrt(2 + t^2)




integrate ln|x^2 - x + 2|




integrate 7/2 / (x^2 - x + 2)



integrate ln |x + 1|



integrate ln |2x + 1|




integrate ln x



Parts twice
integrate e^(2x) cos(3x)




integrate e^(2x) cos(3x)




integrate x^2 cos(pi x)




integrate x^2 cos(pi x)




integrate e^x sin(2x)



integrate e^x sin(2x)




integrate x^2 cos(x)



integrate x^2 cos(x)



integrate x^2 sin(3x + 1) integrate x^2 sin(3x + 1)





question: OCR GCE Maths 4724/01 q2

Parts thrice
integrate (2x^3 + 5x + 1) e^(2x)


integrate x^3 sin(x)









Separation of variables
separation of variables



solve dF/dt = -2F



solve dP/dt = kP



solve dy/dx = 2y



solve (100 - t^2) dN/dt + 4tn = 0



solve dh/dt = (6 - h)/20



solve dy/dx = Cy








Integrating factor
integrating factor

integrating factor




dy/dx = 2x(1 + x^2 - y)




dy/dx = y/x + y^2/x^2



x dy/dx - 4y = 0



d theta / dt + theta/10 = 5 - 5t/2
d theta / dt + theta/10 = 5 - 5t/2




solve x^2 y' + xy = x e^x



solve dy/dx - y / (x + 1) = x - 1



solve dy/dx - y / (x + 1) = x - 1



solve dy/dx + 3x^2y = 6x^2



solve dy/dx + 3x^2y = 6x^2



solve dy/dx + 3x^2y = 6x^2








Fundamental theorem of calculus


differentiate integral from a to x



differentiate integral from 1 to x^2



differentiate integral from a to x



differentiate integral from x to a



differentiate integral from a to x



differentiate integral from a to x



differentiate integral from a to function of x



differentiate integral from a to function of x



fundamental theorem



fundamental theorem



fundamental theorem








Miscellaneous


marginal profit



Cauchy-Reimann equations from continuity of second partials

from left equation from right equation

first result second result




third derivative




iterated integral



iterated integral




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