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

 

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).

Thus the derivative of the given function f(x) = 3x² + ²⁄ₓ + 7 - 4tan(x) + 5sec(x) is 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.

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