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Session 1: Differentiation

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This figure consists of two copies of the complex plane side by side, both focused on the upper-right quadrant. The axes are not labelled. On the left-hand diagram there are two points in the upper-right quadrant, alpha marked with a solid black dot and z marked with a solid blue dot. The point z lies to the right of the point alpha and a little below it. The vector from alpha to z is drawn as a solid blue line with an arrowhead. Between the left-hand and right-hand diagrams is a curved arrow pointing from left to right and labelled f. On the right-hand diagram, three points are marked with solid dots. The first, which is the closest to the origin, is drawn in black labelled f of alpha. The second, also in black, lies above and to the right of the first point, and is labelled f of z. The third point, drawn in blue, lies to the right and a little below the point labelled f of alpha. It is not labelled. Two vectors are also shown, both starting from the point f of alpha. The first vector, drawn in blue, runs from the point f of alpha to the unlabelled blue point. It is an identical copy of the vector from alpha to z that was drawn in the left-hand diagram. The second vector, drawn in black, starts at the point f of alpha and finishes at the point f of z. Above this second vector is the text, scale by the modulus of f prime of alpha. Between the two vectors is a curved arrow, pointing in the anticlockwise direction. Next to it is the text, rotate by Arg f prime of alpha.