If y = (2x + 1)^3, find dy/dx at x = 1.

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Question: If y = (2x + 1)^3, find dy/dx at x = 1.

Options:

Correct Answer: 12

Solution:

dy/dx = 6(2x + 1)^2. At x = 1, dy/dx = 6(2(1) + 1)^2 = 6(3)^2 = 54.

If y = (2x + 1)^3, find dy/dx at x = 1.

If y = (2x + 1)^3, find dy/dx at x = 1.

Correct Answer: 54

Step 1: Start with the equation y = (2x + 1)^3.
Step 2: To find dy/dx, we need to use the chain rule of differentiation.
Step 3: The chain rule states that if y = (u)^n, then dy/dx = n * (u)^(n-1) * (du/dx).
Step 4: Here, let u = (2x + 1) and n = 3.
Step 5: First, find du/dx. Since u = 2x + 1, then du/dx = 2.
Step 6: Now apply the chain rule: dy/dx = 3 * (2x + 1)^(3-1) * (du/dx).
Step 7: Substitute du/dx into the equation: dy/dx = 3 * (2x + 1)^2 * 2.
Step 8: Simplify the equation: dy/dx = 6 * (2x + 1)^2.
Step 9: Now, we need to find dy/dx at x = 1. Substitute x = 1 into the equation.
Step 10: Calculate: dy/dx = 6 * (2(1) + 1)^2.
Step 11: Simplify: dy/dx = 6 * (2 + 1)^2 = 6 * (3)^2.
Step 12: Calculate (3)^2 = 9, so dy/dx = 6 * 9.
Step 13: Finally, calculate 6 * 9 = 54.

Chain Rule – The question tests the application of the chain rule in differentiation, as the function is a composition of functions.
Polynomial Differentiation – It also involves differentiating a polynomial raised to a power, which requires understanding of power rules.