Find the derivative of f(x) = 5x^4 - 3x + 2.
Correct Answer: f'(x) = 20x^3 - 3
- Step 1: Identify the function f(x) = 5x^4 - 3x + 2.
- Step 2: Recognize that we need to find the derivative, which tells us the rate of change of the function.
- Step 3: Use the power rule for derivatives, which states that if f(x) = ax^n, then f'(x) = n * ax^(n-1).
- Step 4: Apply the power rule to the first term, 5x^4. The derivative is 4 * 5x^(4-1) = 20x^3.
- Step 5: Apply the power rule to the second term, -3x. This is the same as -3x^1, so the derivative is 1 * -3x^(1-1) = -3.
- Step 6: The third term, 2, is a constant, and the derivative of a constant is 0.
- Step 7: Combine the derivatives from Steps 4, 5, and 6. So, f'(x) = 20x^3 - 3 + 0.
- Step 8: Simplify the expression to get the final derivative: f'(x) = 20x^3 - 3.
- Power Rule – The power rule states that the derivative of x^n is n*x^(n-1).
- Constant Rule – The derivative of a constant is zero.
