Math Problem Statement

The image contains several tasks asking to find the derivative of different functions. The three tasks are as follows: 1. Task 1: Find the derivative of the following functions: f(x) = 5x - 4, f(x) = 4x^3, f(x) = 3x^2 - 8x, f(x) = x^6 - 5x, f(x) = 3x^2 + 8x - 10. 2. Task 2: Find the derivative of the following functions: f(x) = (x + 4)(5x - 4), f(x) = (x - 3)(4x^3), f(x) = (x^2 - 3)(2x^2 - 10x), f(x) = (3x^3 + 2x)(x^6 - 4x), f(x) = (2x^2 + 8x - 5)(3x^2 + 8x). 3. Task 3: Find the derivative of the following functions: f(x) = (5x - 4) / (2x), f(x) = 4x / (8x - 9), f(x) = 3x - 8 + 9x - 4x, f(x) = sqrt(6x - 10), f(x) = (x^2 - 3x)^2.

Solution

The image you uploaded contains a set of exercises for calculating derivatives of various functions. I will break down the problems for you and assist in finding the solutions.

Let's go through each task in the image:

Задание 1 (Task 1)

Find the derivative of the following functions:

  1. f(x)=5x4f(x) = 5x - 4
  2. f(x)=4x3f(x) = 4x^3
  3. f(x)=3x28xf(x) = 3x^2 - 8x
  4. f(x)=x65xf(x) = x^6 - 5x
  5. f(x)=3x2+8x10f(x) = 3x^2 + 8x - 10

Задание 2 (Task 2)

Find the derivative of the following functions:

  1. f(x)=(x+4)(5x4)f(x) = (x + 4)(5x - 4)
  2. f(x)=(x3)(4x3)f(x) = (x - 3)(4x^3)
  3. f(x)=(x23)(2x210x)f(x) = (x^2 - 3)(2x^2 - 10x)
  4. f(x)=(3x3+2x)(x64x)f(x) = (3x^3 + 2x)(x^6 - 4x)
  5. f(x)=(2x2+8x5)(3x2+8x)f(x) = (2x^2 + 8x - 5)(3x^2 + 8x)

Задание 3 (Task 3)

Find the derivative of the following functions:

  1. f(x)=5x42xf(x) = \frac{5x - 4}{2x}
  2. f(x)=4x8x9f(x) = \frac{4x}{8x - 9}
  3. f(x)=3x8+9x4xf(x) = 3x - 8 + 9x - 4x
  4. f(x)=6x10f(x) = \sqrt{6x - 10}
  5. f(x)=(x23x)2f(x) = (x^2 - 3x)^2

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Product Rule
Quotient Rule
Chain Rule

Formulas

d/dx[x^n] = n*x^(n-1)
Product Rule: d/dx[u(x) * v(x)] = u'(x) * v(x) + u(x) * v'(x)
Quotient Rule: d/dx[u(x) / v(x)] = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Theorems

Power Rule
Product Rule
Quotient Rule
Chain Rule

Suitable Grade Level

Grades 10-12