Find the derivative of f(x) = ln(x^2 + 1).
Correct Answer: (2x)/(x^2 + 1)
- Step 1: Identify the function you want to differentiate, which is f(x) = ln(x^2 + 1).
- Step 2: Recognize that you need to use the chain rule because you have a function inside another function (ln and x^2 + 1).
- Step 3: The chain rule states that if you have a function g(h(x)), the derivative is g'(h(x)) * h'(x).
- Step 4: In this case, g(u) = ln(u) where u = x^2 + 1. First, find the derivative of g(u). The derivative of ln(u) is 1/u.
- Step 5: Now, find the derivative of h(x) = x^2 + 1. The derivative h'(x) = 2x.
- Step 6: Apply the chain rule: f'(x) = g'(h(x)) * h'(x) = (1/(x^2 + 1)) * (2x).
- Step 7: Simplify the expression: f'(x) = 2x / (x^2 + 1).
- Chain Rule – The chain rule is used to differentiate composite functions, which in this case involves the natural logarithm and a polynomial.
- Logarithmic Differentiation – Understanding how to differentiate logarithmic functions, particularly the natural logarithm, is essential.
- Quotient Rule – While not directly used here, recognizing when to apply the quotient rule can be a related concept when dealing with derivatives.
