October 29, 2025

What Is The Derivative Of Fx 3x 2x 1 And What Does It Represent

Q: How do you verify the derivative calculation?

Verify by applying the limit definition: f'(x) = lim_{h→0} [f(x+h) - f(x)] / h, which expands to lim_{h→0} [3(x+h)² + 2(x+h) - 1 - (3x² + 2x - 1)] / h = 6x + 2, matching the power rule result.

Q: What does f'(x) = 6x + 2 tell us about the function?

It indicates the function is increasing where f'(x) > 0 (x > -1/3), decreasing where f'(x) < 0 (x < -1/3), with a minimum at x = -1/3, revealing concavity and critical points.

Q: Can this derivative be used for optimization?

Yes, set f'(x) = 0 to find 6x + 2 = 0, so x = -1/3. The second derivative f''(x) = 6 > 0 confirms a local minimum, useful for minimizing costs or maximizing profits in applied scenarios.

Q: Is the derivative only for polynomials like this?

No, derivatives apply to many functions, but for polynomials, the power rule simplifies it. Common misconception: constants don't affect the derivative, as they represent fixed values without change.

Learn how to find the derivative of f(x) = 3x² + 2x - 1 using power rules, calculate it step-by-step, and understand its meaning as the instantaneous rate of change in calculus.

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Finding the Derivative

The derivative of f(x) = 3x² + 2x - 1 is found using the power rule and basic differentiation rules. For 3x², the derivative is 6x; for 2x, it is 2; and the constant -1 has a derivative of 0. Thus, f'(x) = 6x + 2.

Key Principles of Differentiation

Differentiation follows these principles: the power rule states that d/dx(x^n) = n x^{n-1}, applied term-by-term for polynomials. Here, the quadratic term 3x² yields 3*2x = 6x, the linear term 2x yields 2, and constants vanish, resulting in a linear function f'(x) = 6x + 2.

Practical Example

Consider f(x) = 3x² + 2x - 1 representing the position of an object in feet after t seconds. The derivative f'(t) = 6t + 2 gives velocity in feet per second. At t=1, velocity is 8 ft/s, showing how the slope of the tangent line at any point on the graph indicates instantaneous speed.

Significance and Applications

The derivative represents the instantaneous rate of change of the function, crucial in physics for velocity and acceleration, economics for marginal costs, and optimization. For this quadratic, it models accelerating motion or parabolic trends, helping predict behavior at specific points.

Frequently Asked Questions

How do you verify the derivative calculation?

What does f'(x) = 6x + 2 tell us about the function?

Can this derivative be used for optimization?

Is the derivative only for polynomials like this?