Math Problem Statement

Differentiate f(x) = (3x + 3)3 using the chain rule

Solution

To differentiate f(x)=(3x+3)3f(x) = (3x + 3)^3 using the chain rule, we follow these steps:

Step 1: Identify the outer and inner functions.

  • The outer function is g(u)=u3g(u) = u^3, where u=3x+3u = 3x + 3.
  • The inner function is h(x)=3x+3h(x) = 3x + 3.

Step 2: Differentiate the outer function.

The derivative of g(u)=u3g(u) = u^3 with respect to uu is: g(u)=3u2g'(u) = 3u^2

Step 3: Differentiate the inner function.

The derivative of h(x)=3x+3h(x) = 3x + 3 with respect to xx is: h(x)=3h'(x) = 3

Step 4: Apply the chain rule.

According to the chain rule, the derivative of f(x)=(3x+3)3f(x) = (3x + 3)^3 is given by: f(x)=g(h(x))h(x)f'(x) = g'(h(x)) \cdot h'(x) Substitute the derivatives from Steps 2 and 3: f(x)=3(3x+3)23f'(x) = 3(3x + 3)^2 \cdot 3

Step 5: Simplify the expression.

f(x)=9(3x+3)2f'(x) = 9(3x + 3)^2

So, the derivative of f(x)=(3x+3)3f(x) = (3x + 3)^3 is: f(x)=9(3x+3)2f'(x) = 9(3x + 3)^2

Would you like more details on any step?

Relative questions:

  1. How does the chain rule work for functions with more than two layers?
  2. What happens when the inner function is a trigonometric function?
  3. Can you differentiate f(x)=(5x2+4)4f(x) = (5x^2 + 4)^4 using the chain rule?
  4. How does the power rule relate to the chain rule?
  5. What is the product rule and how does it differ from the chain rule?

Tip:

When differentiating composite functions, always start by identifying the inner and outer functions clearly to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Chain Rule
Composite Functions

Formulas

f'(x) = g'(h(x)) * h'(x)
g'(u) = 3u^2
h'(x) = 3

Theorems

Chain Rule

Suitable Grade Level

Grades 11-12 (Advanced Calculus)