Understanding the Second Derivative of f(x) √(3x^2 - 2x - 1)
Calculus is a powerful tool in mathematics that helps us understand the behavior of functions. One essential concept in calculus is the second derivative, which provides information about the concavity of a function and its rates of change. Let's explore how to find the second derivative of the function f(x) √(3x^2 - 2x - 1) and clarify any ambiguities in the given expressions.
The Function
Consider the function y √(3x^2 - 2x - 1). This function can be rewritten by first squaring both sides to eliminate the square root:
y^2 3x^2 - 2x - 1
Finding the First Derivative
To find the first derivative, we differentiate both sides with respect to x:
2yy' 6x - 2
Divide both sides by 2y to isolate y':
y' frac{6x - 2}{2y} frac{3x - 1}{y}
Finding the Second Derivative
Now, we find the second derivative by differentiating y' with respect to x. Let's use the quotient rule, which states:
[frac{d}{dx}left(frac{u}{v}right) frac{vfrac{du}{dx} - ufrac{dv}{dx}}{v^2}]
In this case, u 3x - 1 and v y. Therefore:
[y'' frac{ycdot3 - (3x - 1)cdotfrac{dy}{dx}}{y^2}]
We already know that y' frac{3x - 1}{y}. Let's substitute this into the expression:
[y'' frac{ycdot3 - (3x - 1)cdotfrac{3x - 1}{y}}{y^2}]
Simplify the numerator:
[y'' frac{3y - frac{(3x - 1)^2}{y}}{y^2}]
Multiply the numerator and denominator by y to clear the fraction:
[y'' frac{3y^2 - (3x - 1)^2}{y^3}]
Now, let's substitute y^2 3x^2 - 2x - 1 back into the equation:
[y'' frac{3(3x^2 - 2x - 1) - (3x - 1)^2}{y^3}]
Simplify the expression in the numerator:
[y'' frac{9x^2 - 6x - 3 - (9x^2 - 6x 1)}{y^3}]
Further simplify:
[y'' frac{9x^2 - 6x - 3 - 9x^2 6x - 1}{y^3} frac{-4}{y^3}]
Conclusion
The second derivative of the function y √(3x^2 - 2x - 1) is:
y'' -frac{4}{y^3}
This result provides valuable insights into the concavity of the function. Understanding the second derivative is crucial for further analysis in calculus and its applications.