Math Problem Statement

Выберите все верные утверждения.

Solution

Let's go through each statement to determine which ones are correct. I will analyze each of them in detail.

  1. y=sinxy = \sin x. Then y(7)=cosxy^{(7)} = \cos x.

    • Since sinx\sin x and cosx\cos x have periodic derivatives, the derivatives of y=sinxy = \sin x repeat every four terms as follows: y=cosx,y=sinx,y=cosx,y(4)=sinx.y' = \cos x, \quad y'' = -\sin x, \quad y''' = -\cos x, \quad y^{(4)} = \sin x.
      • Therefore, y(7)=y(3)=cosxy^{(7)} = y^{(3)} = -\cos x, which does not equal cosx\cos x. This statement is incorrect.
  2. y=2x5y = 2x^5. Then y(7)=0y^{(7)} = 0.

    • y=2x5y = 2x^5 implies that y(6)=0y^{(6)} = 0, so any higher derivatives, including y(7)y^{(7)}, are also zero. This statement is correct.
  3. y=6x3+2xy = 6x^3 + 2x. Then d3y=36dx3d^3y = 36dx^3.

    • y=18x2+2y' = 18x^2 + 2, y=36xy'' = 36x, and y=36y''' = 36.
    • The third derivative is y=36y''' = 36, not 36dx336dx^3. This statement is incorrect.
  4. y=3x4y = 3x^4. Then y=72xy''' = 72x.

    • y=3x4y=12x3y=36x2y=72xy = 3x^4 \Rightarrow y' = 12x^3 \Rightarrow y'' = 36x^2 \Rightarrow y''' = 72x.
    • This statement is correct.
  5. y=axy = a^x. Then d6y=axln6adx6d^6y = a^x \ln^6 a \, dx^6.

    • For y=axy = a^x, each derivative introduces a factor of lna\ln a.
    • y(6)=ax(lna)6y^{(6)} = a^x (\ln a)^6, so d6y=axln6adx6d^6y = a^x \ln^6 a \, dx^6. This statement is correct.
  6. y=2x9+4x4y = 2x^9 + 4x^4. Then y(5)=1008x6y^{(5)} = 1008x^6.

    • y(5)=d5dx5(2x9)=1008x4y^{(5)} = \frac{d^5}{dx^5}(2x^9) = 1008x^4 and for (4x

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Math Problem Analysis

Mathematical Concepts

Differentiation
Higher Order Derivatives
Trigonometric Functions
Polynomial Functions
Exponential Functions

Formulas

Derivative of trigonometric functions: sin(x), cos(x)
Derivative of power functions: d/dx (x^n) = n*x^(n-1)
Exponential function differentiation: d/dx (a^x) = a^x ln(a)

Theorems

Rules for Differentiation
Periodicity of Trigonometric Functions

Suitable Grade Level

University level (Calculus)