# 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.

#### Sum/Difference Rule:

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

#### Derivative of sin(x):

The derivative of sin(x) with respect to x is cos(x).

#### Derivative of cos(x):

The derivative of cos(x) with respect to x is -sin(x).

### Now, let's proceed to find the derivative of 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1 step by step.

#### Step 1: Differentiate the Polynomial Terms

First, let's differentiate the polynomial terms individually:
• Derivative of 4x⁵: Using the power rule, the derivative of 4x⁵ with respect to x is 4 * 5x^(5-1) = 20x⁴.

• Derivative of -7x²: Similarly, the derivative of -7x² is -7 * 2x^(2-1) = -14x.

#### Step 2: Differentiate the Trigonometric Terms

Now, let's differentiate the trigonometric terms individually:
• Derivative of 2sin(x): The derivative of 2sin(x) with respect to x is 2cos(x).

• Derivative of -4cos(x): The derivative of -4cos(x) with respect to x is -4(-sin(x)) = 4sin(x).

#### Step 3: 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) = (20x⁴) - (14x) + (2cos(x)) + (4sin(x)).

#### Step 4: Simplify the Derivative

To simplify the derivative further, we have:
• f'(x) = 20x⁴ - 14x + 2cos(x) + 4sin(x).
Thus the derivative of the given function f(x) = 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1 is 20x⁴ - 14x + 2cos(x) + 4sin(x).

## Conclusion

In calculus, finding the derivative of a function is a fundamental skill that provides critical information about how the function changes with respect to its independent variable. By applying basic differentiation rules and recognizing the derivatives of trigonometric functions, we successfully found the derivative of the composite function f(x) = 4x⁵ - 7x² + 2sin(x) - 4cos(x) - 1. This derivative, f'(x) = 20x⁴ - 14x + 2cos(x) + 4sin(x), is essential for understanding the rate of change of the original function and its behavior as x varies. The process demonstrated in this article is a fundamental concept in calculus, applicable in various fields of science and engineering.