If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.

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Question: If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.

Options:

Correct Answer: 22

Solution:

dy/dx = 20x^3 + 6x - 1. At x = 1, dy/dx = 20(1)^3 + 6(1) - 1 = 20 + 6 - 1 = 25.

If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.

If y = 5x^4 + 3x^2 - x, find dy/dx at x = 1.

Correct Answer: 25

Step 1: Identify the function y = 5x^4 + 3x^2 - x.
Step 2: Differentiate the function with respect to x to find dy/dx.
Step 3: Apply the power rule to each term: The derivative of 5x^4 is 20x^3, the derivative of 3x^2 is 6x, and the derivative of -x is -1.
Step 4: Combine the derivatives to get dy/dx = 20x^3 + 6x - 1.
Step 5: Substitute x = 1 into the derivative: dy/dx = 20(1)^3 + 6(1) - 1.
Step 6: Calculate the values: 20(1) + 6(1) - 1 = 20 + 6 - 1.
Step 7: Simplify the expression: 20 + 6 - 1 = 25.

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 how to differentiate polynomial expressions, which involves applying the power rule.
Evaluation of Derivatives – Substituting a specific value into the derivative to find the slope of the tangent line at that point.