Math Problem Statement
Given the function y, equals, 3, cosine, left parenthesis, square root of, 6, minus, 4, x, cubed, end square root, right parenthesis, commay=3cos(6−4x3), find start fraction, d, y, divided by, d, x, end fraction, .dxdy
Solution
To find the derivative of the given function with respect to , we apply the chain rule. Let's go step by step:
Step 1: Outer function derivative
The outer function is , where . The derivative of with respect to is:
Step 2: Middle function derivative
The middle function is , which can be rewritten as . Its derivative with respect to is: [ \frac{d}{dx}\left[(6 - 4x^3)^{1/2}\right] = \frac{1}{2}(6 - 4x^3)^{-1/2} \cdot \frac{d}{dx}[6 - 4x^3]. ]
Step 3: Inner function derivative
The inner function is . Its derivative with respect to is:
Step 4: Combine the results
Now, substitute the results step by step:
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From Step 3: Substitute into Step 2: Simplify:
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From Step 1: Substitute and : Substitute and :
Step 5: Final simplification
Simplify the expression:
Final Answer:
Would you like further clarification or examples?
Related Questions:
- How does the chain rule work in similar problems involving trigonometric functions?
- What is the interpretation of this derivative in a physical or geometric context?
- How can we simplify for small values of ?
- How does the derivative behave as ?
- Can you apply the product rule alongside the chain rule in more complex expressions?
Tip:
Always rewrite expressions in their simplest forms before differentiating to avoid unnecessary complexity.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Chain Rule
Trigonometric Functions
Formulas
Derivative of cos(u): d/dx[cos(u)] = -sin(u) * du/dx
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Power rule: d/dx[x^n] = n*x^(n-1)
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12 or Introductory College Calculus