Math Problem Statement

Differentiate the following functions: 1) y = (x^3 - 2x)^2, 2) y = x / (x + 1), 3) y = (1 + 2x)^(2/3), 4) y = sqrt(x^3 / (1 - x^2)), 5) y = (x + 2)(2 - 1)^3.

Solution

The image you uploaded contains five problems involving differentiation. Here's a breakdown of each problem along with their instructions:

  1. y=(x32x)2y = (x^3 - 2x)^2
    Task: Find yy'' (second derivative).

  2. y=xx+1y = \frac{x}{x+1}
    Task: Find yy''.

  3. y=(1+2x)23y = \sqrt[3]{(1 + 2x)^2}
    Task: Find yy' (first derivative) and yy'' (second derivative).

  4. y=x31x2y = \sqrt{\frac{x^3}{1 - x^2}}
    Task: Find yy' and yy''.

  5. y=(x+2)(21)3y = (x + 2)(2 - 1)^3
    Task: Find dydx\frac{dy}{dx} and d2ydx2\frac{d^2y}{dx^2}.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Chain Rule
Product Rule
Quotient Rule

Formulas

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

Theorems

Power Rule: d/dx [x^n] = n*x^(n-1)
Derivative of square root and cube root functions

Suitable Grade Level

Undergraduate Calculus