Derivative of 3x³ - 2x² + x - 5
Published on August 12, 2024
Question
f(x) = 3x^3 - 2x^2 + x - 5 f'(x) = ?
Answer
Here’s how to find the derivative of the function, f(x) = 3x³ - 2x² + x - 5:
Understanding Derivatives
The derivative of a function, denoted by f’(x) or df/dx, represents the instantaneous rate of change of the function with respect to its input (x). In simpler terms, it tells us how much the function’s output changes with a tiny change in the input.
Power Rule
The most common rule we use to find derivatives is the power rule. It states:
- d(x^n)/dx = n*x^(n-1)
This means we bring the exponent down as a coefficient and reduce the original exponent by 1.
Applying the Rules
Let’s differentiate f(x) term by term:
-
Term 1: 3x³
- Apply the power rule: (3 * 3)x^(3-1) = 9x²
-
Term 2: -2x²
- Apply the power rule: (-2 * 2)x^(2-1) = -4x
-
Term 3: x
- Remember that x is the same as x¹. Applying the power rule: (1 * 1)x^(1-1) = 1
-
Term 4: -5
- The derivative of a constant (a number without any x) is always 0.
Putting it Together
Now, we combine the derivatives of each term to get the derivative of the entire function:
f’(x) = 9x² - 4x + 1
Therefore, the derivative of f(x) = 3x³ - 2x² + x - 5 is f’(x) = 9x² - 4x + 1.