Math Problem Statement

s/(s^2+a^2)^3

Solution

We are asked to differentiate the function:

f(s)=s(s2+a2)3f(s) = \frac{s}{(s^2 + a^2)^3}

To find dds(s(s2+a2)3)\frac{d}{ds}\left( \frac{s}{(s^2 + a^2)^3} \right), we will apply the quotient rule. The quotient rule for differentiation states:

ddx(u(x)v(x))=v(x)u(x)u(x)v(x)(v(x))2\frac{d}{dx}\left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}

Step 1: Identify u(s)u(s) and v(s)v(s)

Here,

  • u(s)=su(s) = s
  • v(s)=(s2+a2)3v(s) = (s^2 + a^2)^3

Step 2: Differentiate u(s)u(s) and v(s)v(s)

  • u(s)=dds(s)=1u'(s) = \frac{d}{ds}(s) = 1
  • v(s)=dds((s2+a2)3)v'(s) = \frac{d}{ds} \left( (s^2 + a^2)^3 \right)

For v(s)v'(s), we apply the chain rule: v(s)=3(s2+a2)2dds(s2+a2)=3(s2+a2)22s=6s(s2+a2)2v'(s) = 3(s^2 + a^2)^2 \cdot \frac{d}{ds}(s^2 + a^2) = 3(s^2 + a^2)^2 \cdot 2s = 6s(s^2 + a^2)^2

Step 3: Apply the quotient rule

Using the quotient rule:

dds(s(s2+a2)3)=(s2+a2)31s6s(s2+a2)2(s2+a2)6\frac{d}{ds}\left( \frac{s}{(s^2 + a^2)^3} \right) = \frac{(s^2 + a^2)^3 \cdot 1 - s \cdot 6s(s^2 + a^2)^2}{(s^2 + a^2)^6}

Step 4: Simplify the expression

Simplify the numerator:

(s2+a2)36s2(s2+a2)2(s^2 + a^2)^3 - 6s^2(s^2 + a^2)^2

Factor out (s2+a2)2(s^2 + a^2)^2 from both terms:

(s2+a2)2[(s2+a2)6s2]=(s2+a2)2(a25s2)(s^2 + a^2)^2 \left[ (s^2 + a^2) - 6s^2 \right] = (s^2 + a^2)^2 (a^2 - 5s^2)

Thus, the derivative is:

(s2+a2)2(a25s2)(s2+a2)6=a25s2(s2+a2)4\frac{(s^2 + a^2)^2 (a^2 - 5s^2)}{(s^2 + a^2)^6} = \frac{a^2 - 5s^2}{(s^2 + a^2)^4}

Final Answer:

dds(s(s2+a2)3)=a25s2(s2+a2)4\frac{d}{ds} \left( \frac{s}{(s^2 + a^2)^3} \right) = \frac{a^2 - 5s^2}{(s^2 + a^2)^4}


Would you like more details, or have any questions?

Here are some related questions:

  1. What are the steps for using the product rule instead of quotient rule?
  2. How can we apply the chain rule to functions with higher powers?
  3. What happens when a=0a = 0 in this function?
  4. How does the quotient rule differ from the chain rule?
  5. Can you find the second derivative of the function?

Tip: In complex derivatives, always simplify each term before attempting further steps.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Quotient Rule
Chain Rule

Formulas

Quotient Rule: d/dx[u(x)/v(x)] = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Theorems

Quotient Rule
Chain Rule

Suitable Grade Level

Undergraduate Calculus