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Answer 1

The derivatives of the functions are listed below:

(a) $f'(x) = -7\cdot x^{-\frac{9}{2} }- 2\cdot x + 4 - \frac{1}{5} - 5\cdot x^{-2}$    

(b) $f'(x) = \frac{1}{3}\cdot (x + 3)^{-\frac{2}{3} }\cdot (x+ 5)^{\frac{1}{3} } + \frac{1}{3} \cdot (x + 5)^{-\frac{2}{3} } \cdot (x + 3)^{\frac{1}{3} }$

(c) f'(x) = [(cos x + sin x) · (x² - 1) - (sin x - cos x) · (2 · x)] / (x² - 1)²    

(d) f'(x) = (5ˣ · ㏑ 5) · ㏒₅ x + 5ˣ · [1 / (x · ㏑ 5)]

(e) f'(x) = 45 · (x⁻⁵ + √3)⁻⁸ · x⁻⁶

(f) $f'(x) = (\ln x + 1)\cdot [7^{x\cdot \ln x \cdot \ln 7}+7\cdot (x\cdot \ln x)^{6}]$

(g) $f'(x) = -2\cdot \arccos x \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} \right) - \left(\frac{1}{1 + x} \right) \cdot \left(\frac{1}{2} \cdot x^{-\frac{1}{2} }\right)$

(h) f'(x) = cot x + cos (㏑ x) · (1 / x)

How to find the first derivative of a group of functions

In this question we must obtain the first derivatives of each expression by applying differentiation rules:

(a) $f(x) = 2 \cdot x^{-\frac{7}{2} } - x^{2} + 4 \cdot x - \frac{x}{5} + \frac{5}{x} - \sqrt[11]{2022}$

1. $f(x) = 2 \cdot x^{-\frac{7}{2} } - x^{2} + 4 \cdot x - \frac{x}{5} + \frac{5}{x} - \sqrt[11]{2022}$        Given
2. $f(x) = 2 \cdot x^{-\frac{7}{2} } - x^{2} + 4\cdot x - \frac{x}{5} + 5 \cdot x^{-1} - \sqrt[11]{2022}$      Definition of power
3. $f'(x) = -7\cdot x^{-\frac{9}{2} }- 2\cdot x + 4 - \frac{1}{5} - 5\cdot x^{-2}$       Derivative of constant and power functions / Derivative of an addition of functions / Result

(b) $f(x) = \sqrt[3]{x + 3} \cdot \sqrt[3]{x + 5}$

1. $f(x) = \sqrt[3]{x + 3} \cdot \sqrt[3]{x + 5}$              Given
2. $f(x) = (x + 3)^{\frac{1}{3} }\cdot (x + 5)^{\frac{1}{3} }$           Definition of power
3. $f'(x) = \frac{1}{3}\cdot (x + 3)^{-\frac{2}{3} }\cdot (x+ 5)^{\frac{1}{3} } + \frac{1}{3} \cdot (x + 5)^{-\frac{2}{3} } \cdot (x + 3)^{\frac{1}{3} }$        Derivative of a product of functions / Derivative of power function / Rule of chain / Result

(c) f(x) = (sin x - cos x) / (x² - 1)

1. f(x) = (sin x - cos x) / (x² - 1)          Given
2. f'(x) = [(cos x + sin x) · (x² - 1) - (sin x - cos x) · (2 · x)] / (x² - 1)²       Derivative of cosine / Derivative of sine / Derivative of power function / Derivative of a constant / Derivative of a division of functions / Result

(d) f(x) = 5ˣ · ㏒₅ x

1. f(x) = 5ˣ · ㏒₅ x             Given
2. f'(x) = (5ˣ · ㏑ 5) · ㏒₅ x + 5ˣ · [1 / (x · ㏑ 5)]       Derivative of an exponential function / Derivative of a logarithmic function / Derivative of a product of functions / Result

(e) f(x) = (x⁻⁵ + √3)⁻⁹

1. f(x) = (x⁻⁵ + √3)⁻⁹          Given
2. f'(x) = - 9 · (x⁻⁵ + √3)⁻⁸ · (- 5) · x⁻⁶       Rule of chain / Derivative of sum of functions / Derivative of power function / Derivative of constant function
3. f'(x) = 45 · (x⁻⁵ + √3)⁻⁸ · x⁻⁶     Associative and commutative properties / Definition of multiplication / Result

(f) $f(x) = 7^{x\cdot \ln x} + (x \cdot \ln x)^{7}$

1. $f(x) = 7^{x\cdot \ln x} + (x \cdot \ln x)^{7}$         Given
2. $f'(x) = 7^{x\cdot\ln x} \cdot \ln 7 \cdot (\ln x + 1) + 7\cdot (x\cdot \ln x)^{6}\cdot (\ln x + 1)$         Rule of chain / Derivative of sum of functions / Derivative of multiplication of functions / Derivative of logarithmic functions / Derivative of potential functions
3. $f'(x) = (\ln x + 1)\cdot [7^{x\cdot \ln x \cdot \ln 7}+7\cdot (x\cdot \ln x)^{6}]$        Distributive property / Result

(g) $f(x) = \arccos^{2} x - \arctan (\sqrt{x})$

1. $f(x) = \arccos^{2} x - \arctan (\sqrt{x})$        Given
2. $f'(x) = -2\cdot \arccos x \cdot \left(\frac{1}{\sqrt{1 - x^{2}}} \right) - \left(\frac{1}{1 + x} \right) \cdot \left(\frac{1}{2} \cdot x^{-\frac{1}{2} }\right)$      Derivative of the subtraction of functions / Derivative of arccosine / Derivative of arctangent / Rule of chain / Derivative of power functions / Result

(h) f(x) = ㏑ (sin x) + sin (㏑ x)

1. f(x) = ㏑ (sin x) + sin (㏑ x)          Given
2. f'(x) = (1 / sin x) · cos x + cos (㏑ x) · (1 / x)        Rule of chain / Derivative of sine / Derivative of natural logarithm /Derivative of addition of functions
3. f'(x) = cot x + cos (㏑ x) · (1 / x)      cot x = cos x / sin x / Result

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