Math Problem Statement

Find the derivative of the function u(x) = (x^2 + 1)^6 x^2 - 1 and f(x) = e^(-x) / (x e^x).

Solution

Let's break down the two problems from the image:

  1. First Function: The function provided is: u(x)=(x2+1)6x2−1u(x) = (x^2 + 1)^6 x^2 - 1 You are asked to find the derivative of u(x)u(x), which involves applying the product rule and chain rule.

  2. Second Function: The second function is: f(x)=e−xxexf(x) = \frac{e^{-x}}{x e^x} To find the derivative of this function, we will use the quotient rule, as we have a fraction.

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Math Problem Analysis

Mathematical Concepts

Derivative
Product Rule
Chain Rule
Quotient Rule

Formulas

Product Rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x)
Quotient Rule: (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x)) / g(x)^2

Theorems

Derivative Theorem
Product Rule
Chain Rule
Quotient Rule

Suitable Grade Level

Grades 11-12

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