Math Problem Statement

Find dy/dx for y = (3x - 2)^3

Solution

From the image you've provided, the problem is asking to find dydx\frac{dy}{dx} where:

y=(3x2)3y = (3x - 2)^3

You have already begun solving this by introducing u=3x2u = 3x - 2, making the expression y=u3y = u^3. Here's how to complete the process step by step:

Step-by-step solution:

  1. Substitute yy in terms of uu: y=u3y = u^3

  2. Differentiate y=u3y = u^3 with respect to uu: dydu=3u2\frac{dy}{du} = 3u^2

  3. Differentiate u=3x2u = 3x - 2 with respect to xx: dudx=3\frac{du}{dx} = 3

  4. Chain Rule: Using the chain rule, dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}: dydx=3u2×3=9u2\frac{dy}{dx} = 3u^2 \times 3 = 9u^2

  5. Substitute back u=3x2u = 3x - 2: dydx=9(3x2)2\frac{dy}{dx} = 9(3x - 2)^2

So, the derivative is:

dydx=9(3x2)2\frac{dy}{dx} = 9(3x - 2)^2

Would you like more details on any of these steps, or do you have any questions?

Related questions:

  1. How do you apply the chain rule in more complex expressions?
  2. Can you explain implicit differentiation and when to use it?
  3. How do derivatives of higher-order polynomials differ from simple ones?
  4. How do you find the second derivative d2ydx2\frac{d^2y}{dx^2} for this function?
  5. What are some practical applications of this type of derivative?

Tip:

When using the chain rule, always check for inner functions that can be substituted to simplify the differentiation process!

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Chain Rule

Formulas

dy/dx = dy/du * du/dx
dy/du = 3u^2
du/dx = 3

Theorems

Chain Rule of Differentiation

Suitable Grade Level

Grades 11-12 (Precalculus/Calculus)