Math Problem Statement

y=3 sin³ (4x²+1)

Solution

The function given is:

y=3sin3(4x2+1)y = 3 \sin^3(4x^2 + 1)

This can be understood as y=3(sin(4x2+1))3y = 3 \left( \sin(4x^2 + 1) \right)^3.

Derivative:

To differentiate this function yy with respect to xx, we apply the chain rule.

  1. Outer function: y=3u3y = 3 u^3, where u=sin(4x2+1)u = \sin(4x^2 + 1).

    • The derivative of 3u33u^3 with respect to uu is 9u29u^2.
  2. Middle function: u=sin(4x2+1)u = \sin(4x^2 + 1).

    • The derivative of sin(v)\sin(v) with respect to vv is cos(v)\cos(v), where v=4x2+1v = 4x^2 + 1.
  3. Inner function: v=4x2+1v = 4x^2 + 1.

    • The derivative of 4x2+14x^2 + 1 with respect to xx is 8x8x.

Now, putting it all together using the chain rule:

dydx=9(sin(4x2+1))2cos(4x2+1)8x\frac{dy}{dx} = 9 \left( \sin(4x^2 + 1) \right)^2 \cdot \cos(4x^2 + 1) \cdot 8x

Simplifying the expression:

dydx=72x(sin(4x2+1))2cos(4x2+1)\frac{dy}{dx} = 72x \left( \sin(4x^2 + 1) \right)^2 \cdot \cos(4x^2 + 1)

This is the derivative of the function.

Do you want details on any step? Here are some related questions to explore:

  1. How do you apply the chain rule to trigonometric functions?
  2. What are the steps to differentiate functions with powers of trigonometric terms?
  3. How does the product rule differ from the chain rule in differentiation?
  4. Can we find the second derivative of the function? How?
  5. 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)