Exercise 1

Differentiate the following functions:

1)  f(x) = 5

2) f(x) = -2x

3) f(x) = -2x + 2

4)f(x) = -2x^2 - 5

5) f(x) = 2x ^4 + x ^3 - x^2 + 4

6) f(x) = \frac{x^3 +2} {3}

7) f(x) = \frac{1} {3x^2}

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Exercise 2

Differentiate the following functions using the power rule:

1) f(x) = \frac{5}{x^5}

2) f(x) = \frac {5}{x ^5} + \frac{3} {x^2}

3) f(x) = \sqrt{x}

4) f(x) = \frac{1}{\sqrt{x}}

5) f(x) = \frac{1}{x\sqrt{x}}

6) f(x) = \sqrt[3]{x^2} + \sqrt{x}

7) f(x) = (x^2 + 3x - 2) ^4

Exercise 3

Differentiate the following functions:

1) f(x) = \sqrt{x^2 - 2x + 3

2) f(x) = \sqrt[3]{\frac{x^2 + 1} {x ^2 - 1}}

Exercise 4

Differentiate the following exponential functions:

1) f(x) = 10 ^ {\sqrt{x}}

2) f(x) = e ^ {3 - x ^2}

3) f(x) = \frac{e^x + e ^ {-x}}{2}

4) f(x) = 3 ^ {2x^2} \cdot \sqrt{x}

5) f(x) = \frac{e ^{2x}}{x ^2}

Exercise 5

Differentiate the following logarithmic functions:

1) f(x) = ln(2x^4 - x^3 + 3x ^2 - 3x)

2) f(x) = ln (\frac{e^x + 1} {e^x - 1})

3) f(x) = log \sqrt{\frac{1+x}{1-x}}

4) f(x) = ln\sqrt{x(1 - x}

5) f(x) = ln \sqrt[3] {\frac{3x}{x + 2}}

6) f(x) = ln \frac{(x - 2) ^3} {\sqrt{2x -1}}

 

Solution of exercise 1

Differentiate the following functions:

1) f(x) = 5

f'(x) = 0

 

2) f(x) = -2x

f'(x) = -2

 

3) f(x) = -2x+2

f'(x) = -2

 

4) f(x) = -2x^2 - 5

f'(x) = -4x

 

5) f(x) = 2x^4 + x^3 - x^2 + 4

f'(x) = 8x^3 + 3x ^2 - 2x

 

6) f(x) = \frac{x ^3 + 2} {3}

f'(x) = x^2

 

7) f(x) = \frac{1} {3x ^2}

f'(x) = \frac{-6x}{(3x^2) ^2} = \frac{-6x} {9x ^4} = -\frac{2}{3x ^3}

 

Solution of exercise 2

Differentiate the following functions using the power rule:

1) f(x) = \frac{5}{x ^5} = 5x ^{-5}

f'(x) = -25 x ^ {-6} = -\frac{25} {x ^6}

 

2) f(x) = \frac{5}{x ^5} + \frac{3} {x ^2} = 5x ^ {-5} + 3x ^ {-2}

f'(x) = -25 x^{-6} - 6x ^{-3} =- \frac {25} {x^6}- \frac{6} {x ^3}

 

3) f(x) = \sqrt{x} = x ^ {\frac{1}{2}}

f'(x) = \frac{1}{2} x ^ {\frac{1}{2} - 1} = \frac {1}{2} x ^ {-\frac{1}{2}} = \frac{1} {2 \sqrt{x}}

 

4) f(x) = \frac{1}{\sqrt{x}} = x ^ {-\frac{1}{2}}

f'(x) = -\frac {1}{2} x ^ {-\frac{1}{2} - 1} =- \frac{1}{2} x ^ {-\frac{3}{2}} = -\frac{1}{2 \sqrt{x ^3}}

 

5) f(x) = \frac{1}{x \sqrt{x}} = \frac{1}{x \cdot x ^{\frac{1}{2}}} = x ^ {-\frac{3}{2}}

f'(x) = \frac{-3}{2} x ^ {\frac{-5}{2}} = -\frac{3}{2 \sqrt{x ^5}}

 

6) f(x) = \sqrt[3]{x ^2} + \sqrt{x} = x ^ {\frac{2}{3}} + x ^ {\frac{1}{2}}

f'(x) = \frac{2}{3} x ^ {\frac{2}{3} - 1} + \frac{1}{2} x ^ {\frac{1}{2} - 1} = \frac{2}{3} x ^ { -\frac{1}{3}} + \frac{1}{2} x ^ {-\frac{1}{2}} = \frac{2}{3 \sqrt[3]{x}} +\frac{1}{2 \sqrt{x}}

 

7) f(x) = (x ^2 + 3x - 2)^4
f'(x) = 4 (x ^2 + 3x - 2)^3 (2x + 3)

Solution of exercise 3

Differentiate the following functions:

1) f(x) = \sqrt{x ^2 -2x +3}

f'(x) = \frac{2x - 2} {2\sqrt{x ^2 - 2x +3}} = \frac {x - 1} {\sqrt{x ^2 - 2x + 3}}

 

2) f(x) = \sqrt[3] {\frac{x^2 + 1} {x ^2 - 1}}
f' (x) = \frac { \frac {2x (x ^2 - 1) - (x^2 + 1) 2x} {(x ^2 - 1) ^2}} {3 \sqrt[3] ({\frac{x ^2 + 1}{x ^2 - 1}}}) ^ 2
= \frac{\frac{-4x} {(x^2 - 1) ^2 }} {3 \sqrt[3] ({\frac {x ^2 + 1} {x ^2 - 1}) ^ 2}}
= \frac {-4x} { 3 (x ^2 - 1) ^ 2 \sqrt [3] {(\frac {x ^2 + 1} { x ^2 - 1}) ^ 2 }}
= \frac {-4x} {3 \sqrt[3] {(\frac{x ^2 + 1} { x^2 - 1}) ^2} (x ^2 - 1) ^ 6}

= \frac {-4x} { 3 \sqrt[3] { (x ^2 + 1) ^2 (x ^2 - 1) ^4}}

= \frac { -4x} {3 \sqrt[3] {(x ^4 - 1) ^2 (x ^2 - 1) ^2}}

Solution of exercise 4

Differentiate the following exponential functions:

1) f(x) = 10 ^ {\sqrt{x}}

f'(x) = \frac{1}{2 \sqrt{x}} \cdot 10 ^ {\sqrt{x}} \cdot ln 10

 

2) f(x) = e ^ {3 - x^2}

f'(x) = -2x \cdot e ^ {3 - x ^2}

 

3) f(x) = \frac {e ^x + e ^ {-x}} {2}

f'(x) = \frac {e ^x - e^ {-x}} {2}

 

4) f(x) = 3 ^ {2x^2} \cdot \sqrt{x}

f'(x) = 4x \cdot 3 ^ {2x ^2} \cdot ln 3 \cdot \sqrt{x} + \frac {3 ^ {2x^2}} { 2 \sqrt{x}}

 

5) f(x) = \frac {e ^ {2x}} { x ^2}

f'(x) = \frac{ 2 \cdot e ^ {2x} \cdot x ^ 2 - e ^ {2x} \cdot 2x} { x ^4} = \frac {2x \cdot e ^ {2x} (x - 1)}{x ^ 4}

= \frac {2 \cdot e ^ {2x} (x - 1) } { x ^ 3}

 

Solution of exercise 5

Differentiate the following logarithmic functions:

1) f(x) = ln (2x ^4 - x ^3 + 3x ^2 - 3x)

f'(x) = \frac{8x ^3 -3x ^2 + 6x - 3} { 2x ^4 - x ^3 + 3x ^2 - 3x}

 

2) f(x) = ln (\frac{e ^x + 1} { e ^x - 1})
f(x) = ln ( e ^x + 1) - ln (e ^x - 1)
f ' (x) = \frac {e ^x } { e ^x + 1} - \frac{e ^x } { e ^x - 1}
= \frac {e ^ {2x} - e^x - e^{2x} - e^x} { (e ^x + 1) ( e ^x - 1)}
=\frac {-2e ^x} {e ^ {2x} - 1}

 

3) f(x) = log \sqrt{ \frac{1 + x} { 1 - x}}
f(x) = \frac{1}{2} [log (1 + x) - log (1 - x)]
f'(x) = \frac {1}{2} (\frac{1}{1 + x} - \frac{-1} {1 - x} ) \cdot log e = \frac {1}{2} \frac{1 - x + 1 + x} {1 - x ^2 } \cdot log e
= \frac {1} { 1 - x ^2} \cdot log e

 

4) f(x) = ln \sqrt{x (1 - x)}
f(x) = \frac {1}{2} [lnx + ln (1 - x)}]
f ' (x) = \frac {1}{2} ( \frac{1}{x} + \frac{-1}{1- x}) = \frac {1}{2} \cdot \frac {1 - x - x} {x ( 1- x)}
= \frac {1 - 2x} {2x (1 - x)}

 

5) f(x) = ln \sqrt[3]{\frac{3x}{x + 2}}

f(x) = \frac{1}{3} [ln3x - ln (x + 2)]
f'(x) = \frac {1}{3} (\frac{3}{3x} - \frac {1}{x + 2}) = \frac{1}{3} \cdot \frac{x + 2 - x} {x (x + 2)}
=\frac{2}{3x (x + 2)}

 

6) f(x) = ln \frac{(x - 2) ^3} {\sqrt{2x - 1}}

f(x) = ln (x - 2) ^3 - ln \sqrt{2x - 1} = 3ln(x - 2) - \frac{1}{2} ln (2x - 1)

f'(x) = 3 \cdot \frac{1} {x - 2} - \frac {1} {2} \cdot \frac{2 } {2x - 1} = \frac {6x - 3 - x + 2} { (x -2) (2x - 1)}

= \frac {5x - 1} { (x - 2) (2x - 1)}

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.