Math Problem Statement

fx of (2x+1)^3

Solution

To find the derivative of the function f(x)=(2x+1)3f(x) = (2x+1)^3, we'll apply the chain rule. Here's the process:

Steps:

  1. Let u=2x+1u = 2x+1, so f(x)=u3f(x) = u^3.
  2. The derivative of u3u^3 with respect to uu is 3u23u^2.
  3. The derivative of u=2x+1u = 2x+1 with respect to xx is 2.

Applying the Chain Rule:

f(x)=ddx[u3]dudx=3u22=6u2f'(x) = \frac{d}{dx}\left[ u^3 \right] \cdot \frac{du}{dx} = 3u^2 \cdot 2 = 6u^2

Substituting u=2x+1u = 2x+1 back: f(x)=6(2x+1)2f'(x) = 6(2x+1)^2

Final Answer:

f(x)=6(2x+1)2f'(x) = 6(2x+1)^2


Would you like me to provide a step-by-step explanation or solve similar examples? Here are 5 related questions for further exploration:

  1. What is the derivative of f(x)=(3x5)4f(x) = (3x-5)^4?
  2. How do you apply the chain rule to f(x)=(x2+1)5f(x) = (x^2 + 1)^5?
  3. Can you compute f(x)f'(x) for f(x)=(x+1)(2x+3)3f(x) = (x+1)(2x+3)^3?
  4. What is the second derivative of f(x)=(2x+1)3f(x) = (2x+1)^3?
  5. How do you expand and differentiate (2x+1)3(2x+1)^3 directly?

Tip: Always simplify the inner and outer derivatives separately when using the chain rule for complex functions.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Chain Rule

Formulas

Derivative of a composite function: f'(x) = g'(h(x)) * h'(x)

Theorems

Chain Rule of Differentiation

Suitable Grade Level

Grades 11-12