1.2 Exercises

From Förberedande kurs i matematik 2

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{{Mall:Ej vald flik|[[1.2 Deriveringsregler|Teori]]}}
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{{Not selected tab|[[1.2 Rules of differentiation|Theory]]}}
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{{Mall:Vald flik|[[1.2 Övningar|Övningar]]}}
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{{Selected tab|[[1.2 Exercises|Examples]]}}
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===Övning 1.2:1===
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===Example 1.2:1===
<div class="ovning">
<div class="ovning">
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Beräkna derivatan av följande funktioner och förenkla svaret så långt som möjligt
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Calculate the derivative of the following functions and write the answer in simplest possible form:
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%"| <math>\displaystyle\frac{x \ln x}{\sin x}</math>
|width="33%"| <math>\displaystyle\frac{x \ln x}{\sin x}</math>
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</div>{{#NAVCONTENT:Svar|Svar 1.2:1|Lösning a|Lösning 1.2:1a|Lösning b|Lösning 1.2:1b|Lösning c|Lösning 1.2:1c|Lösning d|Lösning 1.2:1d|Lösning e|Lösning 1.2:1e|Lösning f|Lösning 1.2:1f}}
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</div>{{#NAVCONTENT:Answer|Answer 1.2:1|Solution a|Solution 1.2:1a|Solution b|Solution 1.2:1b|Solution c|Solution 1.2:1c|Solution d|Solution 1.2:1d|Solution e|Solution 1.2:1e|Solution f|Solution 1.2:1f}}
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===Övning 1.2:2===
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===Example 1.2:2===
<div class="ovning">
<div class="ovning">
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Beräkna derivatan av följande funktioner och förenkla svaret så långt som möjligt
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Calculate the derivative of the following functions and write the answer in simplest possible form:
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%"| <math>\cos \sqrt{1-x}</math>
|width="33%"| <math>\cos \sqrt{1-x}</math>
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</div>{{#NAVCONTENT:Svar|Svar 1.2:2|Lösning a|Lösning 1.2:2a|Lösning b|Lösning 1.2:2b|Lösning c|Lösning 1.2:2c|Lösning d|Lösning 1.2:2d|Lösning e|Lösning 1.2:2e|Lösning f|Lösning 1.2:2f}}
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</div>{{#NAVCONTENT:Answer|Answer 1.2:2|Solution a|Solution 1.2:2a|Solution b|Solution 1.2:2b|Solution c|Solution 1.2:2c|Solution d|Solution 1.2:2d|Solution e|Solution 1.2:2e|Solution f|Solution 1.2:2f}}
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===Övning 1.2:3===
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===Example 1.2:3===
<div class="ovning">
<div class="ovning">
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Beräkna derivatan av följande funktioner och förenkla svaret så långt som möjligt
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Calculate the derivative of the following functions and write the answer in simplest possible form:
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="33%"| <math> \ln (\sqrt{x} + \sqrt{x+1})</math>
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|width="33%"| <math> \ln (\sqrt{x} + \sqrt{x+1}\,)</math>
|b)
|b)
|width="33%"| <math>\sqrt{\displaystyle \frac{x+1}{x-1}}</math>
|width="33%"| <math>\sqrt{\displaystyle \frac{x+1}{x-1}}</math>
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|width="33%"| <math>x^{\tan x}</math>
|width="33%"| <math>x^{\tan x}</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 1.2:3|Lösning a|Lösning 1.2:3a|Lösning b|Lösning 1.2:3b|Lösning c|Lösning 1.2:3c|Lösning d|Lösning 1.2:3d|Lösning e|Lösning 1.2:3e|Lösning f|Lösning 1.2:3f}}
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</div>{{#NAVCONTENT:Answer|Answer 1.2:3|Solution a|Solution 1.2:3a|Solution b|Solution 1.2:3b|Solution c|Solution 1.2:3c|Solution d|Solution 1.2:3d|Solution e|Solution 1.2:3e|Solution f|Solution 1.2:3f}}
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===Example 1.2:4===
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Calculate the second derivative of the following functions and write the answer in simplest possible form:
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{| width="100%" cellspacing="10px"
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|a)
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|width="50%"| <math>\displaystyle\frac{x}{\sqrt{1-x^2}}</math>
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|b)
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|width="50%"| <math>x ( \sin \ln x +\cos \ln x )</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 1.2:4|Solution a|Solution 1.2:4a|Solution b|Solution 1.2:4b}}

Current revision

       Theory          Examples      

Example 1.2:1

Calculate the derivative of the following functions and write the answer in simplest possible form:

a) \displaystyle \cos x \cdot \sin x b) \displaystyle x^2\ln x c) \displaystyle \displaystyle\frac{x^2+1}{x+1}
d) \displaystyle \displaystyle\frac{\sin x}{x} e) \displaystyle \displaystyle\frac{x}{\ln x} f) \displaystyle \displaystyle\frac{x \ln x}{\sin x}
Answer
Answer 1.2:1
Solution a
Solution 1.2:1a
Solution b
Solution 1.2:1b
Solution c
Solution 1.2:1c
Solution d
Solution 1.2:1d
Solution e
Solution 1.2:1e
Solution f
Solution 1.2:1f

Example 1.2:2

Calculate the derivative of the following functions and write the answer in simplest possible form:

a) \displaystyle \sin x^2 b) \displaystyle e^{x^2+x} c) \displaystyle \sqrt{\cos x}
d) \displaystyle \ln \ln x e) \displaystyle x(2x+1)^4 f) \displaystyle \cos \sqrt{1-x}
Answer
Answer 1.2:2
Solution a
Solution 1.2:2a
Solution b
Solution 1.2:2b
Solution c
Solution 1.2:2c
Solution d
Solution 1.2:2d
Solution e
Solution 1.2:2e
Solution f
Solution 1.2:2f

Example 1.2:3

Calculate the derivative of the following functions and write the answer in simplest possible form:

a) \displaystyle \ln (\sqrt{x} + \sqrt{x+1}\,) b) \displaystyle \sqrt{\displaystyle \frac{x+1}{x-1}} c) \displaystyle \displaystyle\frac{1}{x\sqrt{1-x^2}}
d) \displaystyle \sin \cos \sin x e) \displaystyle e^{\sin x^2} f) \displaystyle x^{\tan x}
Answer
Answer 1.2:3
Solution a
Solution 1.2:3a
Solution b
Solution 1.2:3b
Solution c
Solution 1.2:3c
Solution d
Solution 1.2:3d
Solution e
Solution 1.2:3e
Solution f
Solution 1.2:3f

Example 1.2:4

Calculate the second derivative of the following functions and write the answer in simplest possible form:

a) \displaystyle \displaystyle\frac{x}{\sqrt{1-x^2}} b) \displaystyle x ( \sin \ln x +\cos \ln x )
Answer
Answer 1.2:4
Solution a
Solution 1.2:4a
Solution b
Solution 1.2:4b
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