Math Problem Statement

Find the derivatives as described in the image, including d^3y/dx^3 at x=2 and dy/dx for given equations.

Solution

The image you uploaded contains two calculus exercises related to finding derivatives. Let's break down the problems:

Exercise 20:
You are asked to find the third derivative, d3ydx3\frac{d^3y}{dx^3}, at x=2x = 2 for two expressions of yy.

  1. y=13x3+12x2+x+1y = \frac{1}{3}x^3 + \frac{1}{2}x^2 + x + 1
  2. y=(1+5x)2y = (1 + 5x)^2

Exercise 21:
You are asked to find dydx\frac{dy}{dx}, where yy is given in two forms, involving a substitution variable uu:

  1. y=11+u2y = \frac{1}{1 + u^2} with u=2x+1u = 2x + 1
  2. y=u+1uy = u + \frac{1}{u} with u=(3x+1)4u = (3x + 1)^4

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Math Problem Analysis

Mathematical Concepts

Derivatives
Higher-order derivatives
Chain Rule
Product Rule

Formulas

d/dx (x^n) = nx^(n-1)
Chain Rule: dy/dx = dy/du * du/dx
Higher-order derivatives: d^3y/dx^3 = d/dx (d^2y/dx^2)

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Calculus or Advanced High School (Grade 11-12)