Math Problem Statement

derivative of x/(x+2)^2

Solution

To differentiate the function f(x)=x(x+2)2f(x) = \frac{x}{(x+2)^2}, we will use the quotient rule.

The quotient rule states: (u(x)v(x))=u(x)v(x)u(x)v(x)(v(x))2\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} where u(x)=xu(x) = x and v(x)=(x+2)2v(x) = (x+2)^2.

Step 1: Differentiate the numerator

u(x)=xu(x)=1u(x) = x \quad \Rightarrow \quad u'(x) = 1

Step 2: Differentiate the denominator

v(x)=(x+2)2v(x)=2(x+2)v(x) = (x+2)^2 \quad \Rightarrow \quad v'(x) = 2(x+2)

Step 3: Apply the quotient rule

f(x)=1(x+2)2x2(x+2)((x+2)2)2f'(x) = \frac{1 \cdot (x+2)^2 - x \cdot 2(x+2)}{((x+2)^2)^2}

Step 4: Simplify the expression

Now simplify the numerator: f(x)=(x+2)22x(x+2)(x+2)4f'(x) = \frac{(x+2)^2 - 2x(x+2)}{(x+2)^4} Expand both terms in the numerator: (x+2)2=x2+4x+4(x+2)^2 = x^2 + 4x + 4 2x(x+2)=2x2+4x2x(x+2) = 2x^2 + 4x

So, the numerator becomes: x2+4x+4(2x2+4x)=x2+4x^2 + 4x + 4 - (2x^2 + 4x) = -x^2 + 4

Thus, the derivative simplifies to: f(x)=x2+4(x+2)4f'(x) = \frac{-x^2 + 4}{(x+2)^4}

Final answer:

f(x)=x2+4(x+2)4f'(x) = \frac{-x^2 + 4}{(x+2)^4}

Would you like more details or have any questions? Here are some related questions:

  1. How do we apply the quotient rule in more complex scenarios?
  2. What happens when the denominator has higher powers?
  3. Can the derivative of a similar function be calculated using logarithmic differentiation?
  4. How does chain rule apply in problems involving higher powers?
  5. 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)