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First step is to understand all the notation, i.e. the prime for derivative. We then need to understand that the derivative is the gradient of the tangent at a point. In this case at the point P we can fi...

The product rule states (assuming x' is the differential of x): (fg)​′​​=f​′​​g+fg​′​​ Substitute the values into the rule: (sin(x)cos(x))' = sin(x)'cos(x) + sin(x)cos(x)' (sin(x)cos(x))'...

I don't know how effectively I can communicate this answer via text without the whiteboard but I'll try.

i) First implicitly differentiate with respect to x: 2x + 2y * dy/dx = 0.

Rearranging...