We have:
int int (x * sin(y)) dx dy
Integrate x first:
int(x)dx = 1/2 * x2 + C
Now integrate sin(y):
int(sin(y))dy = -cos(y) + C
Multiply:
-1/2 * x2 * cos(y) + C
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dy/dx = 3 integral = (3x^2)/2
By using the chain rule: dy/dx = dy/du x du/dx With y = tan2x Let u = tan x Then: y = u2 du/dx = d/dx tan x = sec2x dy/dx = dy/du x du/dx = 2u sec2x = 2 tan x sec2x
By the 'Chain Rule' dy/dx = dy/du X du/dx Y = Cos (pi*x). Let pi*x = u Y = Cos(u) u = pi*x dy/du = -Sin(u) du/dx = pi Hence dy/dx = dy/du X du/dx => ( Chain Rule) dy/dx = -Sin(u) X pi Substitute u for pi*x Hence dy/dx = -Sin(pi*x) X pi Tidying up dy/dx = -piSin(pix) Done!!!!
y=x3+ 2x, dx/dt=5, x=2, dy/dt=? Differentiate the equation with respect to t. dy/dt=3x2*dx/dt Substitute in known values. dy/dt=3(2)2 * (5) dy/dt=60
y=2 sin(3x) dy/dx = 2 cos(3x) (3) dy/dx = 6 cos(3x)