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.
To find the derivative of this product of two functions, we will use the product rule, a fundamental rule of differentiation.
Understanding the Product Rule
The product rule states that the derivative of the product of two functions u(x) and v(x) is given by:
- (uv)' = u'v + uv', Where u' and v' represent the derivatives of u(x) and v(x), respectively.
Now, let's apply the product rule to find the derivative of the given function, f(x) = (x² - 2x + 4)(x - 1).
Step 1: Identify the Two Functions
In our case, u(x) = x² - 2x + 4 and v(x) = x - 1.
Step 2: Find the Derivatives of the Two Functions
Now, let's find the derivatives of u(x) and v(x).
Derivative of u(x):
To find the derivative of u(x) = x² - 2x + 4 with respect to x, we'll apply the power rule:
- u'(x) = 2x - 2.
Derivative of v(x):
To find the derivative of v(x) = x - 1 with respect to x, we apply the power rule again:
- v'(x) = 1.
Step 3: Apply the Product Rule
Now that we have found the derivatives of both functions, we can apply the product rule to find the derivative of their product, f(x):
- f'(x) = (u'v) + (uv').
Using the product rule, we get:
- f'(x) = [(2x - 2)(x - 1)] + [(x² - 2x + 4)(1)].
Now, let's simplify this expression.
Step 4: Simplify the Derivative
To simplify the derivative further, we will distribute and combine like terms:
- f'(x) = (2x² - 2x - 2x + 2) + (x² - 2x + 4).
Now, combine like terms:
- f'(x) = (2x² - 4x + 2) + (x² - 2x + 4).
Now, combine like terms again:
- f'(x) = 3x² - 6x + 6.
Thus the derivative of (x² - 2x + 4)(x - 1) is 3x² - 6x + 6.
Conclusion
In calculus, finding the derivative of a function is a fundamental skill that provides valuable information about how the function changes concerning its independent variable. In this article, we used the product rule, a key differentiation technique, to find the derivative of the function f(x) = (x² - 2x + 4)(x - 1). However, it's important to note that the simplified derivative we obtained is f'(x) = 3x² - 6x + 6, not 3(x² - 2x + 2). This derivative represents the rate of change of the original function and is essential for various applications in mathematics, science, and engineering. Understanding and applying differentiation rules like the product rule are crucial for analyzing complex relationships between variables and modeling real-world phenomena.
