If y = 4x^3 - 2x + 1, find dy/dx at x = -1.

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Question: If y = 4x^3 - 2x + 1, find dy/dx at x = -1.

Options:

Correct Answer: -6

Solution:

dy/dx = 12x^2 - 2. At x = -1, dy/dx = 12(-1)^2 - 2 = 12 - 2 = 10.

If y = 4x^3 - 2x + 1, find dy/dx at x = -1.

If y = 4x^3 - 2x + 1, find dy/dx at x = -1.

Correct Answer: 10

Step 1: Identify the function given in the question, which is y = 4x^3 - 2x + 1.
Step 2: Find the derivative of the function y with respect to x, which is denoted as dy/dx.
Step 3: To find the derivative, apply the power rule: for each term, multiply the coefficient by the exponent and decrease the exponent by 1.
Step 4: Differentiate each term: The derivative of 4x^3 is 12x^2, the derivative of -2x is -2, and the derivative of the constant 1 is 0.
Step 5: Combine the derivatives to get dy/dx = 12x^2 - 2.
Step 6: Now, substitute x = -1 into the derivative to find dy/dx at that point.
Step 7: Calculate dy/dx at x = -1: dy/dx = 12(-1)^2 - 2.
Step 8: Simplify the calculation: (-1)^2 is 1, so 12(1) - 2 = 12 - 2.
Step 9: Finally, calculate 12 - 2 to get the result, which is 10.

Differentiation – The process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable.
Polynomial Functions – Understanding the properties and behavior of polynomial functions, including their derivatives.
Evaluation of Derivatives – Calculating the value of the derivative at a specific point.