# 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 derivative of the sum or difference of two functions is the sum or difference of their derivatives, respectively.

Derivative of tan(x): The derivative of tan(x) with respect to x is sec²(x).

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 3x² + ²⁄ₓ + 7 - 4tax(x) + 5sec(x) step by step.

#### Step 1: Differentiate the Polynomial Terms

First, let's differentiate the polynomial terms individually:
• Derivative of 3x²: Using the power rule, the derivative of 3x² with respect to x is 3 * 2x^(2-1) = 6x.

#### Step 2: Differentiate the Fractional Term

Now, let's differentiate the fractional term, ²⁄ₓ.
• Derivative of ²⁄ₓ: The derivative of ²⁄ₓ with respect to x can be computed by applying the power rule for x^(-1):
• Derivative = (-1) * ²/x² = -2/x².

#### Step 3: Differentiate the Trigonometric Terms

Now, let's differentiate the trigonometric terms individually:
• Derivative of -4tan(x): The derivative of -4tan(x) with respect to x is -4 * sec²(x).
• Derivative of 5sec(x): The derivative of 5sec(x) with respect to x is 5 * 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) = (6x) + (-2/x²) + 0 + (-4sec²(x)) + (5sec(x)tan(x)).

#### Step 5: Simplify the Derivative

To simplify the derivative further, we have:
• f'(x) = 6x - 2/x² - 4sec²(x) + 5sec(x)tan(x).

### Conclusion

Calculus, particularly finding the derivative of a function, is a fundamental concept that plays a crucial role in various scientific and engineering fields. By applying basic differentiation rules and recognizing the derivatives of trigonometric functions, we successfully found the derivative of the composite function f(x) = 3x² + ²⁄ₓ + 7 - 4tan(x) + 5sec(x). This derivative, f'(x) = 6x - 2/x² - 4sec²(x) + 5sec(x)tan(x), provides valuable insights into the rate of change of the original function as x varies. The process outlined in this article is an essential concept in calculus, applicable in various fields where understanding changing quantities is paramount.