Simplify the difference quotient f(x+h) - f(x) all over h.

(1) y = 4x - x^2

(2) f(x)= 2x + 1 over x + 3

y = 4x - x^{2}

f(x+h) = 4(x+h)-(x+h)^{2}

f(x+h)-f(x) = 4(x+h)-(x^{2}+2hx+h^{2})- 4x +x^{2} = 4h -2hx-h^{2}

[f(x+h)-f(x)]/h = 4-2x-h

f(x) = (2x+1)/(x+3)

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

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

=[ (2x+2h+1)(x+3) –(2x+1)(x+h+3)]/[(x+h+3)(x+3)]

=[ (2x+1)(x+3) +2h(x+3) –(2x+1)(x+3) -h(2x+1)]/[(x+h+3)(x+3)]

= h(2x+6-2x-1)/ [(x+h+3)(x+3)]

= 5h/[(x+h+3)(x+3)]

[f(x+h)-f(x)]/h = 5/[(x+h+3)(x+3)]

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