How to find derivative of (x² - 2x + 4)(x - 1) ?
In this article, we will explore the process of finding the derivative of a product of two functions using both the product rule and simplification. Specifically, we will walk through the steps to find the derivative of the function: f(x) = (x² - 2x + 4)(x - 1). Calculus, a branch of mathematics that studies rates of change and accumulation, is a fundamental tool in various scientific, engineering, and mathematical disciplines. A core concept in calculus is finding the derivative of a function, which provides information about how the function changes concerning its independent variable.
Understanding the Product Rule
- (uv)' = u'v + uv', Where u' and v' represent the derivatives of u(x) and v(x), respectively.
Step 1: Identify the Two Functions
Step 2: Find the Derivatives of the Two Functions
Derivative of u(x):
- u'(x) = 2x - 2.
Derivative of v(x):
- v'(x) = 1.
Step 3: Apply the Product Rule
- f'(x) = (u'v) + (uv').
- f'(x) = [(2x - 2)(x - 1)] + [(x² - 2x + 4)(1)].
Step 4: Simplify the Derivative
- f'(x) = (2x² - 2x - 2x + 2) + (x² - 2x + 4).
- f'(x) = (2x² - 4x + 2) + (x² - 2x + 4).
- f'(x) = 3x² - 6x + 6.