What does delta sign d mean?

Answer 1

A delta δ is used to indicate a small change in a variable.

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In calculus it "mutates" into a 'd' to represent differentiation which occurs when the small change is over another small change and both tend to zero. For example, when calculating the slope of curve at a point, a small chord is drawn from that point (x, y) to a nearby point (x + δx, y + δy) also on the curve; the slope of this line is δy/δx and approximates to the slope of the curve at (x, y) - the closer (x + δx, y + δy) is to (x, y), ie the smaller δx (and thus δy) is, the closer the slope of the chord is to the slope at the point itself. As δx→0, δy→0 the slope of the chord tends to 0/0 at the point itself; but each point of the curve has a slope and this "mutates" into dy/dx and becomes differentiation (of the curve at that point).

eg Consider the quadratic y = ax² + bx + c

The slope at point (x, y) can be found by considering the chord to (x + δx, y + δy) and letting δx→0.

y + δy = a(x + δx)² + b(x + δx) + c

→ δy = a(x + δx)² + b(x + δx) + c - y

= a(x² + 2xδx + δx²) + b(x + δx) + c - (ax² + bx + c)

= ax² + 2axδx + aδx² + bx + bδx + c - ax² - bx - c

= 2axδx + bδx + aδx²

= δx(2ax + b + aδx)

→ slope chord = δy/δx = δx(2ax + b + aδx)/δx = 2ax + b + aδx (as δx ≠ 0)

→ The slope at (x, y) = 2ax + b + aδx as δx→0.

Now let δx = 0

→ slope at (x, y) = dy/dx = 2ax + b + a.0 = 2ax + b