Math Problem Statement
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, , at for two expressions of .
Exercise 21:
You are asked to find , where is given in two forms, involving a substitution variable :
- with
- with
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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)
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