# 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 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.

#### Derivative of √x:

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

### With these rules in mind, we can proceed to find the derivative of the x⁴ - 3x² - 7x + 4√x - 5 ? step by step.

#### Step 1: Differentiate the Polynomial Terms

Let's break down the function f(x) = x⁴ - 3x² - 7x into its individual terms and find their derivatives.

• Derivative of x⁴: Using the power rule, the derivative of x⁴ with respect to x is 4*x^(4-1) = 4x³.
• Derivative of -3x²: Similarly, the derivative of -3x² is -3*2x^(2-1) = -6x.
• Derivative of -7x: The derivative of -7x is -7.

#### Step 2: Differentiate the Square Root Term

Now, let's find the derivative of the square root term, 4√x.

Derivative of 4√x: Using the chain rule, where the derivative of √u with respect to u is (1/2) * u^(-1/2), and u = x, we have:

• Derivative = 4 * (1/2) * x^(-1/2) = 2x^(-1/2).

#### 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) = (4x³) - (6x) - 7 + (2x^(-1/2)).

#### Step 4: Simplify the Derivative

To simplify further, let's express all the terms in a common form:

• f'(x) = 4x³ - 6x + 2x^(-1/2) - 7.

Thus the derivative of the function f(x) = x⁴ - 3x² - 7x + 4√x - 5 is 4x³ - 6x + 2/x^(-1/2) - 7

## Conclusion

In calculus, finding the derivative of a function is a fundamental skill. By applying basic differentiation rules and the chain rule when dealing with square root terms, we successfully found the derivative of the given function. The derivative, f'(x) = 4x³ - 6x + 2x^(-1/2) - 7, provides valuable information about the rate of change of the original function f(x). This process is essential for understanding the behavior of functions and is a fundamental concept in calculus.