Derivative Calculation: Differentiating Quadratic Functions

In calculus, understanding the derivative of a function is essential for analyzing the rate of change and behavior of the function. In this article, we will explore the derivative of various quadratic functions. Specifically, we will walk through the detailed steps to find the derivative of the function fx 2x^2 - 3x - 1, fx 3x^2 - 2x - 1, and fx x^2 - 2x - 3, and provide a thorough explanation of the process using the chain rule and power rule.

Derivative of fx 2x^2 - 3x - 1

The function in question is fx 2x^2 - 3x - 1. To find its derivative, we will use the power rule of differentiation. The power rule states that if fx ax^n, then df/dx anx^(n-1).

Step-by-step solution:

Given: fx 2x^2 - 3x - 1

Apply the power rule:

- For the term 2x^2, the derivative is 4x.

- For the term -3x, the derivative is -3.

- For the constant term -1, the derivative is 0.

Combine the results: df/dx 4x - 3

Therefore, the derivative of fx 2x^2 - 3x - 1 with respect to x is 4x - 3.

Derivative of fx 3x^2 - 2x - 1 with Respect to x

Next, let's consider the function fx 3x^2 - 2x - 1. We will differentiate this function using the chain rule and then simplify the expression.

Step-by-step solution:

Let ysub 3x^2 - 2x - 1.

Set fy y^(1/2)

For the inner function y 3x^2 - 2x - 1:

- The derivative of 3x^2 is 6x

- The derivative of -2x is -2

- The derivative of the constant term -1 is 0

Apply the chain rule:

df/dx (1/2)y^(-1/2) * (6x - 2)

Simplify and substitute back y:

df/dx (3x - 1)/sqrt(3x^2 - 2x - 1)

So, the derivative of fx 3x^2 - 2x - 1 with respect to x is (3x - 1)/sqrt(3x^2 - 2x - 1).

Derivative of fx x^2 - 2x - 3

This is a straightforward application of the power rule to a quadratic function. Let's walk through the differentiation:

Given: fx x^2 - 2x - 3

For the term x^2, the derivative is 2x;

For the term -2x, the derivative is -2;

For the constant term -3, the derivative is 0.

Assemble the results: df/dx 2x - 2. So, the derivative of fx x^2 - 2x - 3 with respect to x is 2x - 2.

Conclusion

Understanding the derivative of quadratic functions is fundamental for many applications in calculus, physics, and engineering. Whether using the power rule for simple differentiation or the chain rule for more complex functions, mastering these concepts is crucial. For further reading, you can explore more problems and tutorials on derivatives and applications in calculus.