Math Problem Statement
Differentiate f(x) = (3x + 3)3 using the chain rule
Solution
To differentiate using the chain rule, we follow these steps:
Step 1: Identify the outer and inner functions.
- The outer function is , where .
- The inner function is .
Step 2: Differentiate the outer function.
The derivative of with respect to is:
Step 3: Differentiate the inner function.
The derivative of with respect to is:
Step 4: Apply the chain rule.
According to the chain rule, the derivative of is given by: Substitute the derivatives from Steps 2 and 3:
Step 5: Simplify the expression.
So, the derivative of is:
Would you like more details on any step?
Relative questions:
- How does the chain rule work for functions with more than two layers?
- What happens when the inner function is a trigonometric function?
- Can you differentiate using the chain rule?
- How does the power rule relate to the chain rule?
- 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)