單元 26: 指數函數的微分

單元 _26: 指數函數的微分

( {… § ^5.4)

欲}析ÖNbƒbDúbƒbíbç模型_, Ûê|l

NbƒbDúbƒbíûƒbíd†_. íl_,

Nbƒbí}d†

dx (e^x) = e^x

<„_> I _{f (x) = e}^x_. 根Wûƒbíì2_, Nb J£

”Ìí4”_,

f⁰(x) = lim

h→0

f (x + h) − f (x) h

= lim

h→0

e^x+h − e^x

h = lim

h→0

e^x(e^h − 1) h

= e^x lim

h→0

e^h − 1

h (1)

QO_, 根W{… ₃₆₁ 頁í[ 4 (Ïí„p超|…z

¸圍_, ôI_), )

lim

h→0

e^h − 1

h = 1

Ĥ_, â _{(1) ,} ) d

dx (e^x) = f⁰(x) = e^x(1) = e^x

Ç_, 根W©鎖d†_, ª)ªƒbDNbƒb¯Aí}

d†, ¹

Nbƒbí©鎖d†

dx

e^{f (x)} = e^{f (x)}f⁰(x)

<„_> qƒb

g(x) = e^x

†¯Aƒb

h(x) ^def= g[f (x)] = e^{f (x)} ]â©鎖d†J£Nbƒbí}d†_, ¹

g⁰(x) = e^x )

d dx

e^{f (x)} = h⁰(x) = g⁰[f (x)]f⁰(x)

= e^{f (x)}f⁰(x)

註 _.

JNb¶}íûƒb_, ¹ d

dx

e^{f (x)} = (ƒb…™₎₍Nb¶}íûƒb₎

= e^{f (x)}f⁰(x)

6ÿuz_, DÀÓíNbƒbí}d†ó°_, øáÑ ƒb…™_, O根W©鎖d†_, .â JNb¶}íûƒb_.

W _1.

(a) f (x) = x²e^x

(b) g(t) = (e^t + 2)^3/2

(c) h(x) = e^2x2+x

(d) y = xe⁻²

√x

(e) k(t) = e^t e^t + e^−t

<j_{> (a)} 根W ¶d†DNbƒbí}d†_, ) f⁰(x) = (x²)⁰e^x + x²(e^x)⁰ = 2xe^x + x²e^x

= xe^x(x + 2)

(b) 根W©鎖d†J£Nbƒbí}d†1“_, ) g⁰(x) = 3

2(e^t + 2)^1/2(e^t + 2)⁰

= 3 2e^t

q

e^t + 2

(c) 根WNbƒbí©鎖d†_,

h⁰(x) = e^2x2+x(2x² + x)⁰ = (4x + 1)e^2x2+x (d) 根W ¶d†DNbƒbí©鎖d†1“_, )

dy

dx = e⁻²

√x + xe⁻²

√x −2 · 1 2√

x

!

= e⁻²

√x(1 − √ x)

(e) 根Wζd†DNbƒbí©鎖d†1“_, ) k⁰(t) = e^t(e^t + e^−t) − e^t(e^t + e^−t(−1))

(e^t + e^−t)²

= e^2t + 1 − e^2t + 1

(e^t + e^−t)² = 2

(e^t + e^−t)²

W _2.

Q(t) = Q₀e^kt

t„¤¾ÊL< _t víAÅ0 (growth rate) Q⁰(t) Dçví¾ _Q(t) A£ª_.

<„_> 根WNbƒbí©鎖d†1“_, )

Q⁰(t) = Q₀e^kt(kt)⁰ = kQ₀e^kt = kQ(t)

¹ _Q⁰_(t) D _Q(t) A£ª_, àF°_.

W _3.

f (x) = e^−x2 í¥曲õ_.

<j_> íl_, 根WNbƒbí©鎖d†_, ) f⁰(x) = −2xe^−x2

y根W ¶d†DNbƒbí©鎖d†1“_, ) f⁰⁰(x) = −2e^−x2 + (−2x)e^−x2(−2x)

= 2e^−x2(2x² − 1)

0©/_. ÄÑ _e^−x2 0£_, ]I _f⁰⁰_{(x) = 0,} ) 2x² − 1 = 0

¹ù¥曲`²bÑ

x = − 1

√2 D _{x = √}¹ 2

QO_, l _f⁰⁰ Ê}’|íú_ä–È,í¯U_, )



−∞, −^√1

2



: f⁰⁰ = (+)(+) = (+), f ,凹



−^√1

2, ^√1

2



: f⁰⁰ = (+)(−) = (−), f -凹

√1

2, ∞



: f⁰⁰ = (+)(+) = (+), f ,凹

àÇý_. ÄѬ _{x = −}√¹

2 D _{x =} ^√1

2 v_, 凹4ÌZ‰

/

f − 1

√2

!

= f 1

√2

!

= e^−1/2

])ù_¥曲õ

− 1

√2, e^−1/2

!

D _√¹

2, e^−1/2

!

W _4.

P (t) = 300, 000e^−0.09t+

√t/2, 0 ≤ t ≤ 10 t°¤.ß gí|7ÛM_.

<j_> íl_, 根WNbƒbí©鎖d†1“_, ) P⁰(t) = 300, 000e^−0.09t+

√t/2 1 4√

t − 0.09

!

0©//Ä _e^−0.09t+

√t/2 0£_, ]I _P⁰_{(t) = 0,} ) 1

4√

t − 0.09 = 0

¹ñøí@äbÑ

t = 1

4(0.09)

!₂

=

 1 0.36

2

= 1

12.96 ≈ 7.72

QO_, l _P⁰ Ê}’|íù_ä–È,í¯U_, )



0, _12.96^{1 }: P⁰ = (+)(+) = (+), P 遞Ó

 ₁

12.96, 10^: P⁰ = (+)(−) = (−), P 遞Á

àÇý_. ¢ _P 0©/_, ]根Wø階ûƒbì¶_, ¤.

ß gÊ _7.72 (|7ÛM

P

 1 0.1296



= 300, 000e

−0.09 0.1296+

q 1 0.1296

 2

≈ 600, 779

根W²t_, )

ø般Nbƒbí}d†

dx (b^x) = (ln b)b^x, b > 0, b 6= 1

<„_> 根WNú逆4£úb _, )²t

b^x = e^{ln bx} = e^{(ln b)x}

QO_, 根WNbƒbí©鎖d†_, ) d

dx (b^x) = e^{(ln b)x}[(ln b)x]⁰

= (ln b)e^{(ln b)x} = (ln b)b^x

6ÿuz_, ø般NbƒbíûƒbYÍNbƒb…™_, OÛÖ JbíAÍúb _{ln b.}

°Ü_, 根W©鎖d†_, )

ø般Nbƒbí©鎖d†

dx

b^{f (x)} = (ln b)b^{f (x)}f⁰(x)

<„_> qƒb

g(x) = b^x

†¯Aƒb

h(x) ^def= g[f (x)] = b^{f (x)} ]根W©鎖d†£ø般Nbƒbí}d†_, ¹

g⁰(x) = (ln b)b^x )

d dx

b^{f (x)} = h⁰(x) = g⁰[f (x)]f⁰(x)

= (ln b)b^{f (x)}f⁰(x)

W _5.

(a) f (x) = 3^x x² + 1

(b) g(x) = 2 1 + 5^−x

(c) h(x) = x²2^−1/x

<j_{> (a)} 根W ¶d†Dø般Nbƒbí}d†_, ) f⁰(x) = (ln 3)3^x(x² + 1) − 3^x(2x)

(x² + 1)²

= 3^x[(x² + 1) ln 3 − 2x]

(x² + 1)²

(b) Z寫1根W©鎖d†£ø般Nbƒbí©鎖d†_, ) g⁰(x) = d

dx

h2(1 + 5^−x)⁻¹ⁱ

= −2

(1 + 5^−x)²(1 + 5^−x)⁰

= −2(ln 5)5^−x(−1)

(1 + 5^−x)² = (2 ln 5)5^−x (1 + 5^−x)² (c) 根W ¶d†£ø般Nbƒbí©鎖d†_, )

h⁰(x) = (2x)2^−1/x + x²(ln 2)2^−1/x

 2 x²



= (2x)2^−1/x + (2 ln 2)2^−1/x

= 2(x + ln 2)2^−1/x