How to find derivative of (x⁴ + 4)(x² - 3) ?
In this article, we will explore the process of finding the derivative of a product of two functions. Specifically, we will walk through the steps to find the derivative of the function: f(x) = (x⁴ + 4)(x² - 3). Calculus, a branch of mathematics that studies rates of change and accumulation, is a fundamental tool in various scientific, engineering, and mathematical disciplines. One of the core concepts in calculus is finding the derivative of a function, which provides valuable information about how the function changes concerning its independent variable.
To find the derivative of this product of two functions, we will apply the product rule, a fundamental rule of differentiation.
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): To find the derivative of u(x) = x⁴ + 4 with respect to x, we'll apply the power rule: u'(x) = 4x³ + 0 = 4x³.
- Derivative of v(x): To find the derivative of v(x) = x² - 3 with respect to x, we apply the power rule again: v'(x) = 2x - 0 = 2x.
Step 3: Apply the Product Rule
- f'(x) = (u'v) + (uv').
- f'(x) = [(4x³)(x² - 3)] + [(x⁴ + 4)(2x)].
Step 4: Simplify the Derivative
- f'(x) = (4x⁵ - 12x³) + (2x⁵ + 8x).
- f'(x) = 4x⁵ + 2x⁵ - 12x³ + 8x.
- f'(x) = 6x⁵ - 12x³ + 8x.