Exploring the Derivative of f(x) 3x3 - 4x - 1 and Its Applications

In calculus, understanding the rate of change of a function is crucial. This article will guide you through the process of finding the derivative of the function f(x) 3x3 - 4x - 1, and how to find its gradient, critical points, and maximum and minimum values. We will also discuss the importance of properly differentiating terms with multiple exponents, such as the ambiguous x1 term.

The Function and Its Derivative

Consider the function f(x) 3x3 - 4x - 1. To find the derivative of f(x) with respect to x, we first need to differentiate each term using the power rule: [frac{d}{dx} [ax^n] anx^{n-1}]

Differentiating Each Term

1. The term 3x3: [frac{d}{dx} [3x^3] 9x^2] 2. The term -4x (which is -4x1): [frac{d}{dx} [-4x] -4] 3. The constant term -1: [frac{d}{dx} [-1] 0]

The Complete Derivative

Combining these results, the derivative of f(x) is: [frac{df}{dx} 9x^2 - 4]

Gradient of the Curve at x 2

To find the gradient, or the value of the derivative, at a specific point, say x 2, we substitute this value into the derivative:

At x 2:

[frac{df}{dx} text{ at } x 2 9(2)^2 - 4 9 times 4 - 4 36 - 4 32]

So, the gradient at x 2 is 32.

Identifying Maximum and Minimum Points

To find the maximum and minimum points, we need to find the critical points where the derivative is either zero or undefined. These points correspond to the local maxima and minima of the function:

Setting the Derivative to Zero:

[frac{df}{dx} 0 Rightarrow 9x^2 - 4 0]

Solving for x:

[Rightarrow 9x^2 4 Rightarrow x^2 frac{4}{9} Rightarrow x pm frac{2}{3}]

We now have two critical points: x 2/3 and x -2/3. To determine whether these points are maximum or minimum, we can use the second derivative test. The second derivative of f(x) is found by differentiating the first derivative:

Second Derivative:

[frac{d^2f}{dx^2} frac{d}{dx} [9x^2 - 4] 18x]

Evaluating the second derivative at the critical points:

At x 2/3:

[frac{d^2f}{dx^2} text{ at } x frac{2}{3} 18 left( frac{2}{3} right) 12 > 0]

A positive second derivative indicates a local minimum.

At x -2/3:

[frac{d^2f}{dx^2} text{ at } x -frac{2}{3} 18 left( -frac{2}{3} right) -12

A negative second derivative indicates a local maximum.

Conclusion

By understanding and applying the rules of differentiation, we can solve for the derivative of a function, calculate its gradient at specific points, and determine its critical points. For the function f(x) 3x3 - 4x - 1, the first derivative is 9x2 - 4, the gradient at x 2 is 32, and the function reaches a local minimum at x 2/3 and a local maximum at x -2/3.

Additional Resources

For a deeper understanding of how to differentiate terms with multiple exponents, such as x1, you may find these resources helpful:

YouTube tutorial: How to Differentiate xx Differential Calculus: Computing Derivatives (Lamar University) Khan Academy: Basic Derivative Rules