Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)

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Question: Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)

Options:

Correct Answer: 5x^4 - 9x^2

Solution:

The derivative f\'(x) = d/dx(x^5 - 3x^3 + 2) = 5x^4 - 9x^2.

Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)

Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)

Step 1: Identify the function f(x) = x^5 - 3x^3 + 2.
Step 2: Recall the power rule for derivatives: if f(x) = x^n, then f'(x) = n*x^(n-1).
Step 3: Apply the power rule to the first term x^5: the derivative is 5*x^(5-1) = 5x^4.
Step 4: Apply the power rule to the second term -3x^3: the derivative is -3*3*x^(3-1) = -9x^2.
Step 5: The derivative of the constant term 2 is 0, since constants do not change.
Step 6: Combine the results from Steps 3, 4, and 5 to get the final derivative: f'(x) = 5x^4 - 9x^2.

Power Rule – The derivative of x^n is n*x^(n-1), applied to each term of the polynomial.
Constant Rule – The derivative of a constant is zero, which applies to the constant term in the polynomial.