Math Problem Statement
y=3 sin³ (4x²+1)
Solution
The function given is:
This can be understood as .
Derivative:
To differentiate this function with respect to , we apply the chain rule.
-
Outer function: , where .
- The derivative of with respect to is .
-
Middle function: .
- The derivative of with respect to is , where .
-
Inner function: .
- The derivative of with respect to is .
Now, putting it all together using the chain rule:
Simplifying the expression:
This is the derivative of the function.
Do you want details on any step? Here are some related questions to explore:
- How do you apply the chain rule to trigonometric functions?
- What are the steps to differentiate functions with powers of trigonometric terms?
- How does the product rule differ from the chain rule in differentiation?
- Can we find the second derivative of the function? How?
- How would you graph this function, and what key points should you consider?
Tip: When differentiating composite functions, always start from the outermost function and work inward using the chain rule.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Differentiation
Chain Rule
Formulas
dy/dx = 72x (sin(4x^2 + 1))^2 * cos(4x^2 + 1)
Theorems
Chain Rule in Differentiation
Trigonometric Differentiation
Suitable Grade Level
Grades 11-12 (or Calculus Level)