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Derivative of ln(x^2 + 1)
Published on October 13, 2024
Calculus
Find the derivative of the function h(x) = ln(x² + 1) using the chain rule. The derivative is h'(x) = 2x / (x² + 1).
Question
Find the derivative of h(x) = ln(x²+1)
Answer
Let’s find the derivative of h(x) = ln(x² + 1) step-by-step.
1. Identify the outer function and the inner function.
- The outer function is the natural logarithm: ln( )
- The inner function is the quadratic expression: x² + 1
2. Apply the chain rule.
The chain rule states:
d/dx [f(g(x))] = f’(g(x)) * g’(x)
d/dx [f(g(x))] = f’(g(x)) * g’(x)
In our case:
- f(u) = ln(u)
- g(x) = x² + 1
3. Find the derivatives of the outer and inner functions.
- f’(u) = 1/u
- g’(x) = 2x
4. Substitute into the chain rule formula.
h’(x) = f’(g(x)) * g’(x)
h’(x) = (1/(x² + 1)) * (2x)
5. Simplify the expression.
h’(x) = 2x / (x² + 1)
Therefore, the derivative of h(x) = ln(x² + 1) is h’(x) = 2x / (x² + 1).