= lim(h->0) [-2xh - h^2 + 4h] / h
= lim(h->0) -2x - h + 4
= -2x - 0 + 4
= -2x + 4
6)
f(x) = 2x^3 + 1, f(x + h) = 2(x + h)^3 + 1
f'(x)
= lim(h->0) [f(x + h) - f(x)] / h
= lim(h->0) [(2(x + h)^3 + 1) - (2x^3 + 1)] / h
= lim(h->0) [2x^3 + 6x^2 h + 6xh^2 + 2h^3 + 1 - 2x^3 - 1] / h
= lim(h->0) [6x^2 h + 6xh^2 + 2h^3] / h
= lim(h->0) 6x^2 + 6xh + 2h^2]
= 6x^2 + (6x)(0) + 2(0^2)
= 6x^2 + 0 + 0
= 6x^2
7)
f(x) = 4 + (4 / x), f(x + h) = 4 + (4 / (x + h))
f'(x)
= lim(h->0) [f(x + h) - f(x)] / h
= lim(h->0) [(4 + (4 / (x + h))) - (4 + (4 / x))] / h
= lim(h->0) [4 + (4 / (x + h)) - 4 - (4 / x)] / h
= lim(h->0) [(4 / (x + h)) - (4 / x)] / h
= lim(h->0) [(4x - 4(x + h)) / (x(x + h))] / h
= lim(h->0) [4x - 4x - 4h] / [(x)(x + h)(h)]
= lim(h->0) [-4h] / [(x)(x + h)(h)]
= lim(h->0) -4 / [x(x + h)]
= -4 / [x(x + 0)]
= -4 / ((x)(x))
= -4 / x^2
8)
f(x) = 5 + 3sqrt(x), f(x + h) = 5 + 3sqrt(x + h)
f'(x)
= lim(h->0) [f(x + h) - f(x)] / h
= lim(h->0) [(5 + 3sqrt(x + h)) - (5 + 3sqrt(x))] / h
= lim(h->0) [5 + 3sqrt(x + h) - 5 - 3sqrt(x)] / h
= lim(h->0) [3sqrt(x + h) - 3sqrt(x)] / h
= lim(h->0) [(3)(sqrt(x + h) - sqrt(x))] / h
= lim(h->0) [(3)(sqrt(x + h) - sqrt(x))(sqrt(x + h) + sqrt(x))] / [(h)(sqrt(x + h) + sqrt(x))]
= lim(h->0) [3((x + h) - x)] / [h(sqrt(x + h) + sqrt(x))]
= lim(h->0) [3h] / [h(sqrt(x + h) + sqrt(x))
= lim(h->0) 3 / (sqrt(x + h) + sqrt(x))
= 3 / (sqrt(x + 0) + sqrt(x))
= 3 / (sqrt(x) + sqrt(x))
= 3 / (2sqrt(x))
9)
f(x) = 10sqrt(x + 5), f(x + h) = 10sqrt(x + h + 5)
f'(x)
= lim(h->0) [f(x + h) - f(x)] / h
= lim(h->0) [10sqrt(x + h + 5) - 10sqrt(x + 5)] / h
= lim(h->0) [(10)(sqrt(x + h + 5) - sqrt(x + 5))] / h
= lim(h->0) [(10)(sqrt(x + h + 5) - sqrt(x + 5))(sqrt(x + h + 5) + sqrt(x + 5))] / [(h)(sqrt(x + h + 5) + sqrt(x + 5))]
= lim(h->0) [10((x + h + 5) - (x + 5))] / [(h)(sqrt(x + h + 5) + sqrt(x + 5))]
= lim(h->0) [10h] / [h(sqrt(x + h + 5) + sqrt(x + 5))]
= lim(h->0) 10 / [sqrt(x + h + 5) + sqrt(x + 5)]
= 10 / [sqrt(x + 5) + sqrt(x + 5)]
= 10 / (2sqrt(x + 5))
= 5 / sqrt(x + 5)
10)
f(x) = 3x / (x + 2), f(x + h) = 3(x + h) / (x + h + 2)
f'(x)
= lim(h->0) [f(x + h) - f(x)] / h
= lim(h->0) [(3(x + h) / (x + h + 2)) - (3x / (x + 2))] / h
= lim(h->0) [(3(x + h)(x + 2) - (3x)(x + h + 2)) / ((x + h + 2)(x + 2))] / h
= lim(h->0) [3(x^2 + xh + 2x + 2h - x^2 - xh - 2x)] / ((h)(x + h + 2)(x + 2))
= lim(h->0) [3(2h)] / ((h)(x + h + 2)(x + 2))
= lim(h->0) 6 / ((x + h + 2)(x + 2))
= 6 / ((x + 2)(x + 2))
= 6 / (x + 2)^2