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

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) = (x⁴ + 4)(x² - 3).

Step 1: Identify the Two Functions

In our case, u(x) = x⁴ + 4 and v(x) = x² - 3.

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) = 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

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) = [(4x³)(x² - 3)] + [(x⁴ + 4)(2x)].

Step 4: Simplify the Derivative

To simplify the derivative further, we will distribute and combine like terms:
  • f'(x) = (4x⁵ - 12x³) + (2x⁵ + 8x).
Now, combine like terms:
  • f'(x) = 4x⁵ + 2x⁵ - 12x³ + 8x.
Combine like terms once more:
  • f'(x) = 6x⁵ - 12x³ + 8x.

Thus derivative of (x⁴ + 4)(x² - 3) is 6x⁵ - 12x³ + 8x.

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) = (x⁴ + 4)(x² - 3). The derivative we obtained is f'(x) = 6x⁵ - 12x³ + 8x. 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.

Popular posts from this blog

How to find derivative of x⁴ - 3x² - 7x + 4√x - 5 ?

Image
  In this article, we will delve into the process of finding the derivative of a polynomial function that includes both traditional polynomial terms and a square root term. Specifically, we will work through the steps to find the derivative of the function f(x) = x⁴ - 3x² - 7x + 4√x - 5. Calculus, a branch of mathematics that explores rates of change and the accumulation of quantities, is an essential tool in various fields, from physics to economics. One fundamental concept in calculus is finding the derivative of a function, which provides information about how the function changes with respect to its independent variable.  Understanding Basic Differentiation Rules Before we tackle the given function, let's review some basic rules of differentiation that will be useful: Power Rule:  The derivative of x^n with respect to x, where n is a constant, is n*x^(n-1). Constant Rule:  The derivative of a constant C (where C is any real number) is 0. Sum/Difference Rule:  The derivative of
Read more

How to find derivative of 3x² + ²⁄ₓ + 7 - 4tax(x) + 5sec(x) ?

Image
  In this article, we will walk through the process of finding the derivative of a function that incorporates polynomial terms, a fractional term, and trigonometric functions. Specifically, we will explore how to find the derivative of the function: f(x) = 3x² + ²⁄ₓ + 7 - 4tan(x) + 5sec(x).  Calculus, a branch of mathematics that explores the concepts of rates of change and accumulation, is a foundational tool in various scientific and engineering fields. Among its essential concepts is finding the derivative of a function, which provides insights into how the function changes concerning its independent variable.  Understanding Basic Differentiation Rules Before we delve into finding the derivative of this complex function, let's review some fundamental rules of differentiation: Power Rule: The derivative of x^n with respect to x, where n is a constant, is n * x^(n-1). Constant Rule: The derivative of a constant C (where C is any real number) is 0. Sum/Difference Rule : The deriv
Read more

How to find derivative of 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1 ?

Image
In this article, we will explore the process of finding the derivative of a composite function that includes polynomial terms, sine, and cosine functions. Specifically, we will work through the steps to find the derivative of the function f(x) = 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1.Calculus, the branch of mathematics that deals with rates of change and accumulation, is a crucial tool in various scientific and engineering disciplines. One of the fundamental concepts in calculus is finding the derivative of a function, which describes how the function's values change as the independent variable changes.  Understanding Basic Differentiation Rules Before we dive into finding the derivative of the given function, let's review some fundamental rules of differentiation that will be essential for our calculations: Power Rule:  The derivative of x^n with respect to x, where n is a constant, is n * x^(n-1). Constant Rule:  The derivative of a constant C (where C is any real number) is 0. Su
Read more