What is the second derivative of (x^3)/(x^2-1)? And how would I accurately graph both the derivative and the second derivative of this function?
Just do a web search for online graphing calculators. There are bunches.
y = x3/(x2-1)
y' = [3x2 * (x2-1) - x3 * 2x]/(x2-1)2
= (x4 - 3x2)/(x2-1)2
y'' = [(4x3 - 6x)(x^2-1)2 - (x4 - 3x2)*2*(x2-1)*2x]/(x2-1)4
= (2x3 + 6x)/(x2-1)3
To find the second derivative of the function f(x) = (x^3)/(x^2-1), we need to follow these steps:
Step 1: Find the first derivative, f'(x).
Step 2: Find the second derivative, f''(x), by differentiating the first derivative.
Step 1: Find the first derivative, f'(x):
To find f'(x), we need to apply the quotient rule. The quotient rule states that for a function u(x)/v(x), the derivative is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2.
Using the quotient rule, let's find f'(x):
f(x) = (x^3)/(x^2-1)
Apply the quotient rule:
f'(x) = [(x^2-1)(3x^2) - (x^3)(2x)] / (x^2-1)^2
Simplifying f'(x):
f'(x) = (3x^4 - 3x^2 - 2x^4) / (x^2-1)^2
f'(x) = (x^4 - 3x^2) / (x^2-1)^2
Step 2: Find the second derivative, f''(x):
To find f''(x), we need to differentiate f'(x) using the power rule.
Differentiating f'(x):
f''(x) = [(4x^3 - 6x)(x^2-1)^2 - (x^4 - 3x^2)(2(x^2-1)(2x))]/(x^2-1)^4
Simplifying f''(x):
f''(x) = [(4x^3 - 6x)(x^2-1)^2 - 2x(x^2-1)(x^4 - 3x^2)] / (x^2-1)^4
f''(x) = (4x^5 - 4x^3 - 6x^3 + 6x - 2x^7 + 6x^5) / (x^2-1)^4
f''(x) = (-2x^7 + 10x^5 - 10x^3 + 6x) / (x^2-1)^4
Now that we have the second derivative, f''(x) = (-2x^7 + 10x^5 - 10x^3 + 6x) / (x^2-1)^4, we can use it to accurately graph both the derivative and the second derivative of this function.
To accurately graph the derivative and the second derivative of a function, follow these steps:
1. Plot the x-intercepts: Find the values of x where the derivative or second derivative is equal to zero (i.e., where the numerator equals zero in our case). Plot these points on the x-axis.
2. Identify vertical asymptotes: Find the values of x where the denominator of the derivative or second derivative is equal to zero. Plot vertical lines at these x-values.
3. Determine the behavior near vertical asymptotes: Determine the behavior of the derivative and second derivative as x approaches the vertical asymptotes from both sides by evaluating the limits.
4. Determine the behavior at x = ±∞: Evaluate the limits of the derivative and second derivative as x approaches positive and negative infinity to determine their behavior at the far ends of the graph.
5. Find critical points: Find the points where the derivative or second derivative changes sign. Plot these points on the graph.
6. Analyze intervals: Determine if the function is increasing or decreasing by examining the signs of the derivative. Also, examine the concavity (positive or negative) by examining the signs of the second derivative.
7. Sketch the graph: Use the information you gathered in steps 1-6 to accurately sketch the graph of the derivative and second derivative.
Following these steps will help you graph both the derivative and the second derivative of the function f(x) = (x^3)/(x^2-1).