Find the second derivative of f(x) = ln(x^2 + 1).
Correct Answer: -2/(x^2 + 1)^2
Steps
Concepts
Step 1: Start with the function f(x) = ln(x^2 + 1). Step 2: Find the first derivative f'(x) using the chain rule. The derivative of ln(u) is (1/u) * (du/dx). Here, u = x^2 + 1, so du/dx = 2x. Step 3: Apply the chain rule: f'(x) = (1/(x^2 + 1)) * (2x) = (2x)/(x^2 + 1). Step 4: Now, find the second derivative f''(x) by differentiating f'(x). Use the quotient rule: if g(x) = (2x) and h(x) = (x^2 + 1), then f''(x) = (g'h - gh')/h^2. Step 5: Calculate g' = 2 and h' = 2x. Step 6: Substitute into the quotient rule: f''(x) = (2(x^2 + 1) - (2x)(2x))/((x^2 + 1)^2). Step 7: Simplify the numerator: 2(x^2 + 1) - 4x^2 = 2 - 2x^2. Step 8: So, f''(x) = (2 - 2x^2)/((x^2 + 1)^2). Step 9: Factor out 2 from the numerator: f''(x) = 2(1 - x^2)/((x^2 + 1)^2). Step 10: The final answer can be simplified to f''(x) = -2/(x^2 + 1)^2.
Differentiation – The process of finding the derivative of a function, which measures how the function changes as its input changes.Chain Rule – A rule for computing the derivative of the composition of two or more functions.Quotient Rule – A method for finding the derivative of a function that is the quotient of two other functions.Second Derivative – The derivative of the first derivative, which provides information about the concavity of the function.
