§ 3.4 The derivative as a rate of change f'(x)=lim(h→0)[f(x+h)-f(x)]/h Δ

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3.4 The derivative as a rate of change (視微分為變化率) 3.5 Chain rule (連鎖律)

3.6 Differentiaing the trigonometric functions (三角函數的微分)

3.7 Implicit differentiation, rational powers (隱函數的微分方式，有理數的次方)

§ 3.4 The derivative as a rate of change

f'(x)=lim(h→0)[f(x+h)-f(x)]/h ^Δ_Δ^y_x 平均變化率

=lim(Δ→0)Δy/Δx=瞬間變化率 of f at x

≜變化率 of f at x=rate of change of f at x.

rate of change=change rate在工程裡頭變化率，就是對它做微分。

eg. 若 f(x)=點 x 的位置，則 f'(x)=點 x 的速度，f'' (x)=點 x 的加速度 Ex:P132(6.7)

§ Thm:(The Chain rule)

If g is diff. at x, and f is diff. at g(x), then f∘g is diff. at x and (f∘g)'(x)=[f(g(x))]'=f'(g(x))⋅g'(x)^(f∘g)'(x)=[f(g(x))]'≠f'(g(x))

口語：f 合成 g 的微分等於 f 的微分帶入 g(x)乘上 g(x)的微分 pf:證明留到高微

cor:If y=f(u) and u=g(x). y 是 u 的變數、u 是 x 的變數

Then dy/dx=dy/du⋅du/dx.

eg.

○¹ y=(u-1)/(u+1), u=x^2. Find dy/dx.

pf:dy/dx=dy/du⋅du/dx=[(u+1)-(u-1)]/(u+1)^2⋅2x=4x/(u+1)^2=4x/(x^2+1)^2

○² d/dx[(x^2-1)^100] x^100 的合成函數，代入x^2-1

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pf:100(x^2-1)^99⋅2x=200x(x^2-1)^99

○³ [1/(x^4+2x+1)^2]'

pf:-2/(x^4+2x+1)^3⋅(4x^3+2) x^-2 的合成函數，代入 x^4+2x+1

○⁴ y=2u/(1-4u), u=(5x^2+1)^4. Find dy/dx.

pf:dy/dx=dy/du⋅du/dx=[2(1-4u)+8u]/(1-4u)^2⋅4(5x^2+1)^3⋅10x=…

○⁵ d/dx[f(x^2+1)]= x的合成函數，代入x^2+1

pf:f'(x^2+1)⋅2x

○⁶ d/dx[f³(x^2+1)]= x^3 的合成函數，代入f(x^2+1)

pf:3f²(x^2+1)⋅f'(x^2+1)⋅2x Ex:P138(5.7.16.24.27.44.45.60)

§ 3.6 Differentiating the trigonometric functions Thm:(sinx)'=cosx, (cosx)'=-sinx.

pf:Let x∈R lim(h→0)sinx= lim(h→0)[sin(x+h)-sin(x)]/h

=lim(h→0)(sinxcosh+cosxsinh-sinx)/h

=lim(h→0)[sinx(cosh-1)+cosxsinh]/h

=lim(h→0)[sinx⋅(cosh-1)/h+cosx⋅sinh/h]

sinx→sinx、(cosh-1)/h→0、cosx→cosx、sinh/h→1

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=sinx⋅0+cosx⋅1=cosx

Thm:(tanx)'=sec²x、(secx)'=secxtanx，(cotx)'=-csc²x、(cscx)'=-cscxcotx pf:(tanx)'=(sinx/cosx)' tanx 本來就定義在 cosx 不為零的地方

=(cos²x+sin²x)/cos²x=sec²x eg.

○¹ [(1-secx)/tanx]'=[(-secxtan²x)-(1-secx)sec² x]/ tan²x

○² d/dx[sec(x^2+1)]=sec(x^2+1)tan(x^2+1)⋅2x

○³ [x^3x-sin(2x^2)]'=3x^2-cos(2x^2)⋅4x

○⁴ d/dx[csc(f(3cosx))]=-csc(f(3cosx))cot(f(3cosx))⋅f'(3cosx)⋅(-3sinx) Ex145(12.24.27.55.56.67)

§ 3.7 Implicit differentiation, rational powers.

eg.

○¹ 3x^3y-4y-2x+sinx=0. Find y'=? y=f(x)

9x^2y+3x^3y'-4y'-2+cosx=0 (3x^3-4)y'=-9x^2y+2-cosx

⇒y'=(-9x^2y+2-cosx)/(3x^3-4)

○² cos(x-y)=(2x+1)^2y^2. Find y'.

pf:-sin(x-y)⋅(1-y')=2(2x+1)⋅2⋅y^2+(2x+1)^2⋅2y⋅y'...化簡

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Thm:Let p,q ∈ℤ q≠0,(x^(p/q))'=(p/g)x^(p/q-1) pf: Let y=x^(p/q).Then y^q=x^p.

qy^(q-1)⋅y'=px^(p-1)

q(x^(p/q))^(q-1)⋅y'= px^(p-1)

y'=(p/q)x^(p-1)(x^(p/q))^(1-q)=(p/q)x^(p-1+p/q-p)=(p/q)x^(p/q-1) eg.

p

q p

xq = x

○¹ (x^(2/3))'=(2/3)x^(-1/3)=

3

2 3 x

○² {√[(secx/(1+x^2)]}'= _{2 2}²

2

1 1 sec tan (1 ) 2 sec

2 sec (1 )

1

x x x x x

x x x

+ −

⋅ +

+

Ex:P150(10.18.32.34.42.48)