Math Problem Statement

Find the derivative of the function h(t) = (t^3 / (t^6 + 9))^2.

Solution

The function to differentiate is:

h(t)=(t3t6+9)2h(t) = \left( \frac{t^3}{t^6 + 9} \right)^2

To find h(t)h'(t), we will use the chain rule along with the quotient rule. Here are the steps:

  1. Chain Rule: First, recognize that the outer function is a square, so we will differentiate (something)2(\text{something})^2 first and then multiply by the derivative of the inside.

    Using the chain rule: ddt(u2)=2ududt\frac{d}{dt} \left( u^2 \right) = 2u \cdot \frac{du}{dt} where u=t3t6+9u = \frac{t^3}{t^6 + 9}.

  2. Quotient Rule: Now, to differentiate the inside part t3t6+9\frac{t^3}{t^6 + 9}, we use the quotient rule: ddt(f(t)g(t))=f(t)g(t)f(t)g(t)(g(t))2\frac{d}{dt} \left( \frac{f(t)}{g(t)} \right) = \frac{f'(t)g(t) - f(t)g'(t)}{(g(t))^2} where f(t)=t3f(t) = t^3 and g(t)=t6+9g(t) = t^6 + 9.

    Let's calculate the necessary derivatives:

    • f(t)=3t2f'(t) = 3t^2
    • g(t)=6t5g'(t) = 6t^5

    Substituting these into the quotient rule: ddt(t3t6+9)=3t2(t6+9)t3(6t5)(t6+9)2\frac{d}{dt} \left( \frac{t^3}{t^6 + 9} \right) = \frac{3t^2(t^6 + 9) - t^3(6t^5)}{(t^6 + 9)^2} Simplifying the numerator: 3t2(t6+9)t3(6t5)=3t8+27t26t8=3t8+27t23t^2(t^6 + 9) - t^3(6t^5) = 3t^8 + 27t^2 - 6t^8 = -3t^8 + 27t^2 So the derivative becomes: 3t8+27t2(t6+9)2\frac{-3t^8 + 27t^2}{(t^6 + 9)^2}

  3. Final derivative: Now applying the chain rule to get h(t)h'(t): h(t)=2(t3t6+9)3t8+27t2(t6+9)2h'(t) = 2 \cdot \left( \frac{t^3}{t^6 + 9} \right) \cdot \frac{-3t^8 + 27t^2}{(t^6 + 9)^2} Simplifying the final expression: h(t)=2t3(3t8+27t2)(t6+9)3h'(t) = \frac{2t^3(-3t^8 + 27t^2)}{(t^6 + 9)^3}

So, the derivative is:

h(t)=2t3(3t8+27t2)(t6+9)3h'(t) = \frac{2t^3(-3t^8 + 27t^2)}{(t^6 + 9)^3}

Would you like a more detailed explanation of any part of this solution?


Related questions:

  1. How does the chain rule work in this context?
  2. Can you explain the quotient rule in more depth?
  3. What would change if the power was different, such as (t3t6+9)3\left( \frac{t^3}{t^6 + 9} \right)^3?
  4. How can the quotient rule be generalized for more complex functions?
  5. What are some common mistakes to avoid when applying the quotient rule?

Tip: Always simplify the numerator and denominator before differentiating to make the process easier.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Chain Rule
Quotient Rule

Formulas

Chain Rule: d/dt (u^n) = n * u^(n-1) * du/dt
Quotient Rule: d/dt (f/g) = (f'g - fg') / g^2

Theorems

Chain Rule
Quotient Rule

Suitable Grade Level

Grades 11-12 (Advanced Calculus)