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Use the four step process to find the slope of the tangent line to the graph at any point:

f(x) = -1/2(x^2)

f(x) = -1/2(x^2)

Step 1
f(x + h)= -1/2(x + h)^2
f(x + h)= -1/2(x^2 + 2xh + h^2)
f(x + h)= -1/2 x^2 - xh - 1/2 h^2

Step 2
f(x + h)-f(x)= -1/2 x^2 - xh - 1/2 h^2 - (-1/2 x^2)

f(x + h)-f(x)= -1/2 x^2 - xh - 1/2 h^2 + 1/2 x^2

f(x + h) - f(x) = -xh - 1/2 h^2
f(x + h) - f(x) = h (-x - 1/2 h)

Step 3
(f(x + h) - f(x))/h = (h(-x - 1/2 h))/h
(f(x + h) - f(x))/h = -x - 1/2 h

Step 4
Evaluate lim h-->0
lim h-->0 = -x - 1/2 (0)
lim h-->0 = -x

Dx(-1/2 x^2) = -x

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