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We know that 1. sin(a+b) = sin(a)cos(b)+sin(b)cos(a) and 2. sin(a-b) = sin(a)cos(b)-sin(b)cos(a) Add equations 1. and 2. sin(a+b)+sin(a-b) = 2sin(a)cos(b)+sin(b)cos(a)-sin(b)cos(a) = 2sin(a)cos(b) Let x=a...

For the line passing through A and B: m = (y2-y1)/(x2-x1) = (-6-4)/(7-3) = -5/2

For the perpendicular line: m = -1/(-5/2) = 2/5 

y - y1 = m*(x - x1)  >>  y - 4 = (2/5)*(x - 3)  >>...

y = (3x⁴ - 18)/x

The gradient of a tangent to a curve is equal to dy/dx 

However, we must simplify this equation before we can differentiate it;

y = 3x³ - 18/x =...

y= x³ +5x² -6x +4

dy/dx = 3x² +10x -6

...

dy/dx = 3x^2