IDZ 6.1 – Option 13. Solutions Ryabushko A.P.

Let's study the differentiation of functions using examples. Let's differentiate the following 14 functions:

1. $f(x) = x^2$

Derivative: $f'(x) = 2x$

1. $f(x) = \sqrt{x}$

Derivative: $f'(x) = \frac{1}{2\sqrt{x}}$

1. $f(x) = \frac{1}{x}$

Derivative: $f'(x) = -\frac{1}{x^2}$

1. $f(x) = e^x$

Derivative: $f'(x) = e^x$

1. $f(x) = \ln(x)$

Derivative: $f'(x) = \frac{1}{x}$

1. $f(x) = \sin(x)$

Derivative: $f'(x) = \cos(x)$

1. $f(x) = \cos(x)$

Derivative: $f'(x) = -\sin(x)$

1. $f(x) = \tan(x)$

Derivative: $f'(x) = \sec^2(x)$

1. $f(x) = \cot(x)$

Derivative: $f'(x) = -\csc^2(x)$

10. $f(x) = \sec(x)$

Derivative: $f'(x) = \sec(x)\tan(x)$

11. $f(x) = \csc(x)$

Derivative: $f'(x) = -\csc(x)\cot(x)$

12. $f(x) = \sinh(x)$

Derivative: $f'(x) = \cosh(x)$

13. $f(x) = \cosh(x)$

Derivative: $f'(x) = \sinh(x)$

14. $f(x) = \tanh(x)$

Derivative: $f'(x) = \operatorname{sech}^2(x)$

Besides these 14 examples, there are many other functions that can also be differentiated. Differentiation is one of the most important operations in mathematics and is used in various fields of science and technology.

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"IDZ 6.1 – Option 13. Decisions of Ryabushko A.P." is a digital product containing solutions to mathematics problems in Microsoft Word 2003 format. The product contains detailed solutions to 14 examples of differentiation of functions, namely:

1. $f(x) = x^2$, $f'(x) = 2x$
2. $f(x) = \sqrt{x}$, $f'(x) = \frac{1}{2\sqrt{x}}$
3. $f(x) = \frac{1}{x}$, $f'(x) = -\frac{1}{x^2}$
4. $f(x) = e^x$, $f'(x) = e^x$
5. $f(x) = \ln(x)$, $f'(x) = \frac{1}{x}$
6. $f(x) = \sin(x)$, $f'(x) = \cos(x)$
7. $f(x) = \cos(x)$, $f'(x) = -\sin(x)$
8. $f(x) = \tan(x)$, $f'(x) = \sec^2(x)$
9. $f(x) = \cot(x)$, $f'(x) = -\csc^2(x)$
10. $f(x) = \sec(x)$, $f'(x) = \sec(x)\tan(x)$
11. $f(x) = \csc(x)$, $f'(x) = -\csc(x)\cot(x)$
12. $f(x) = \sinh(x)$, $f'(x) = \cosh(x)$
13. $f(x) = \cosh(x)$, $f'(x) = \sinh(x)$
14. $f(x) = \tanh(x)$, $f'(x) = \operatorname{sech}^2(x)$

The solutions were compiled by the author Ryabushko A.P. and are presented in a convenient html format, which ensures ease of use. Each solution is presented in a separate block with a detailed description of the steps and calculations.

This product can be useful for schoolchildren, students and teachers who study mathematics and want to improve their knowledge and skills in solving problems in differential calculus. Purchasing this product will significantly reduce the time spent preparing for lessons, exams or tests, as well as gain a more complete and in-depth understanding of mathematical concepts.

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