微分公式(5)
Author: 藍邦偉　老師

pf

此結果在n≥0時已證明，若n<0，則令m=-n

令f(x) =x^n=x^{-m}=\frac{1}{x^m}

$⇒f'(x)=\frac{d}{dx} \left( \frac{1}{x^m}  \right) $

$ =\frac{0\cdot x^m-mx^{m-1}\cdot 1}{(x^m)²} $

$ =\frac{-mx^{m-1}}{x^{2m}} $

=-mx^{-m-1}

$ 微分公式(5) $

$ =nx^n $

$ 設y=x^n，n=\frac{q}{p} ，p,q∈\mathbb{Z}，p≠0$

⇒y=x^{\frac{q}{p}}⇒y^p=x^q

$ ⇒\frac{d}{dx} (y^p)=\frac{d}{dx}(x^q)$

$ ⇒py^{p-1}\frac{dy}{dx}=qx^{q-1} $

⇒\frac{dy}{dx}=\frac{qx^{1-1}}{py^{p-1}}

$ =\frac{q}{p}\cdot\frac{x^{q-1}}{(x^{\frac{q}{p}})^{p-1}} $

$ =\frac{q}{p}\cdot \frac{x^{q-1}}{x^{q-\frac{q}{p}}} $

$ =\frac{q}{p}\cdot x^{(q-1)-(q-\frac{q}{p})} $

$ =\frac{q}{p}\cdot x^{\frac{q}{p}-1} $

$ =nx^{n-1} $

n = 2 (slider from 1 to 2, step 1)
Step = 0 (slider from 0 to 7, step 1)
Question: "f(x)=x^n⇒f'(x)=nx^{n-1},n∈" + (LaTeX[If[n ≟ 1, "Z", "Q"]]) + ""
step = 0 (slider from 0 to 10, step 1)
a = If(n ≟ 1, Step ≟ 1)
b = If(n ≟ 1, Step ≟ 1)
c = If(n ≟ 1, Step ≟ 1)
d = If(n ≟ 1, Step ≟ 1)
e = If(n ≟ 1, Step ≟ 1)
f = If(n ≟ 2, Step ≟ 1)
g = If(n ≟ 1, Step ≟ 1)
h = If(n ≟ 2, Step ≟ 1)
i = If(n ≟ 1, Step ≟ 1)
j = If(n ≟ 2, Step ≟ 1)
k = If(n ≟ 1, Step ≟ 1)
l = If(n ≟ 2, Step ≟ 1)
m = If(n ≟ 1, Step ≟ 1)
o = If(n ≟ 2, Step ≟ 1)
p = If(n ≟ 1, step ≟ 1)
q = If(n ≟ 2, step ≟ 1)
r = If(n ≟ 1, step ≟ 1)
s = If(n ≟ 2, step ≟ 1)
t = If(n ≟ 1, step ≟ 1)
u = If(n ≟ 2, step ≟ 1)
v = If(n ≟ 1, step ≟ 1)
w = If(n ≟ 2, step ≟ 1)
z = If(n ≟ 1, step ≟ 1)
a_1 = If(n ≟ 2, step ≟ 1)
b_1 = If(n ≟ 1, Step ≟ 0)
c_1 = If(n ≟ 2, step ≟ 0)
d_1 = If(n ≟ 1, Step ≟ 0)
e_1 = If(n ≟ 2, step ≟ 0)
f_1 = If(n ≟ 1, Step ≟ 0)
g_1 = If(n ≟ 2, step ≟ 0)
h_1 = If(n ≟ 1, Step ≟ 0)
i_1 = If(n ≟ 2, step ≟ 0)
j_1 = If(n ≟ 1, Step ≟ 0)
k_1 = If(n ≟ 2, step ≟ 0)
l_1 = If(n ≟ 1, Step ≟ 0)
m_1 = If(n ≟ 2, step ≟ 0)
n_1 = If(n ≟ 1, Step ≟ 0)
o_1 = If(n ≟ 2, step ≟ 0)