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

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

Options:

Correct Answer: 16

Solution:

dy/dx = 20x^3 - 9x^2 + 2. At x = 1, dy/dx = 20 - 9 + 2 = 13.

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

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

Correct Answer: 13

Step 1: Identify the function given, which is y = 5x^4 - 3x^3 + 2x - 1.
Step 2: Differentiate the function with respect to x to find dy/dx. Use the power rule for each term.
Step 3: Apply the power rule: For 5x^4, the derivative is 20x^3; for -3x^3, the derivative is -9x^2; for 2x, the derivative is 2; and for -1, the derivative is 0.
Step 4: Combine the derivatives to get dy/dx = 20x^3 - 9x^2 + 2.
Step 5: Substitute x = 1 into the derivative: dy/dx = 20(1)^3 - 9(1)^2 + 2.
Step 6: Calculate the values: 20(1) = 20, -9(1) = -9, so dy/dx = 20 - 9 + 2.
Step 7: Simplify the expression: 20 - 9 = 11, then 11 + 2 = 13.
Step 8: Conclude that dy/dx at x = 1 is 13.

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 functions, which are functions that can be expressed in the form of a sum of terms consisting of a variable raised to a non-negative integer power.
Evaluation of Derivatives – Substituting a specific value into the derivative to find the slope of the tangent line at that point.