Math Problem Statement
derivative of x/(x+2)^2
Solution
To differentiate the function , we will use the quotient rule.
The quotient rule states: where and .
Step 1: Differentiate the numerator
Step 2: Differentiate the denominator
Step 3: Apply the quotient rule
Step 4: Simplify the expression
Now simplify the numerator: Expand both terms in the numerator:
So, the numerator becomes:
Thus, the derivative simplifies to:
Final answer:
Would you like more details or have any questions? Here are some related questions:
- How do we apply the quotient rule in more complex scenarios?
- What happens when the denominator has higher powers?
- Can the derivative of a similar function be calculated using logarithmic differentiation?
- How does chain rule apply in problems involving higher powers?
- What are some real-life applications of derivatives in such rational functions?
Tip: Always simplify the expression after applying the quotient rule to avoid errors in further calculations.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Quotient Rule
Simplification
Formulas
Quotient rule: (u(x)/v(x))' = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2
Simplification of algebraic expressions
Theorems
Quotient Rule for Derivatives
Suitable Grade Level
College Calculus or Advanced High School Calculus (Grades 11-12)