# In this article, we will explore the process of finding the derivative of a complex function that combines polynomial terms, radical terms, and trigonometric functions. Specifically, we will demonstrate how to find the derivative of the function: f(x) = 7x⁵ - x⁴ + 4x³/² + 8/√x - 4sec(x). Calculus, the mathematical study of rates of change and accumulation, is a critical tool in various scientific and engineering disciplines. One of its core concepts is finding the derivative of a function, which reveals how the function's values change concerning its independent variable.

## Understanding Basic Differentiation Rules

Before we embark on finding the derivative of this multifaceted 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 derivative of the sum or difference of two functions is the sum or difference of their derivatives, respectively.

Derivative of √x: The derivative of √x with respect to x is (1/2) * x^(-1/2).

Derivative of sec(x): The derivative of sec(x) with respect to x is sec(x) * tan(x).

### Now, let's proceed to find the derivative of  7x⁵ - x⁴ + 4x³/² + 8/√x - 4sec(x) step by step.

#### Step 1: Differentiate the Polynomial Terms

Let's differentiate the polynomial terms individually:
• Derivative of 7x⁵: Using the power rule, the derivative of 7x⁵ with respect to x is 5 * 7x^(5-1) = 35x⁴.
• Derivative of -x⁴: Similarly, the derivative of -x⁴ is -4x^(4-1) = -4x³.
• Derivative of 4x³/²: The derivative of 4x³/² can be computed using the power rule for fractional exponents: Derivative = (3/2) * 4x^(3/2-1) = 6x^(1/2) = 6√x.

#### Step 2: Differentiate the Radical Term

Now, let's differentiate the radical term, 8/√x:
• Derivative of 8/√x: Applying the power rule for radicals, the derivative is -4x^(-1/2) = -4/√x.

#### Step 3: Differentiate the Trigonometric Term

Lastly, let's differentiate the trigonometric term, -4sec(x):
• Derivative of -4sec(x): The derivative of -4sec(x) with respect to x is -4 * sec(x) * tan(x).

#### Step 4: Combine the Derivatives

Now that we have found the derivatives of all the individual terms, we can combine them to find the derivative of the entire function f(x).
• f'(x) = 35x⁴ - 4x³ + 6√x - 4/√x - 4sec(x)tan(x).

#### Step 5: Simplify the Derivative

To simplify the derivative further, we have:
• f'(x) = 35x⁴ - 4x³ + 6√x - 4/√x - 4sec(x)tan(x).

### Conclusion

Calculus, particularly the process of finding the derivative of a function, is a foundational concept in mathematics that plays a crucial role in various scientific and engineering disciplines. By applying basic differentiation rules and understanding the derivatives of radical and trigonometric functions, we successfully found the derivative of the complex function f(x) = 7x⁵ - x⁴ + 4x³/² + 8/√x - 4sec(x). The derivative, f'(x) = 35x⁴ - 4x³ + 6√x - 4/√x - 4sec(x)tan(x), provides valuable insights into how the original function changes as the independent variable x varies. This process exemplifies the essential role of calculus in analyzing and understanding dynamic relationships between variables in a wide range of applications.