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Mathematics, five units · Ages 17–18

Natural Logarithm Derivatives and Domain Practice

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For each item, determine the domain restriction for the function and then compute its first derivative with respect to x.

  1. Find the domain and derivative of f(x) = ln(x).

  2. Find the domain and derivative of f(x) = ln(7x).

  3. Find the domain and derivative of f(x) = ln(x - 5).

  4. Find the domain and derivative of f(x) = ln(x^2 + 3).

  5. Differentiate f(x) = ln(x^4) using logarithm laws first.

  6. Find the derivative of f(x) = x * ln(x).

  7. Find the derivative of f(x) = ln((2x + 1)/(x - 3)).

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Natural Logarithm Derivatives and Domain Practice

Answer key

  1. Find the domain and derivative of f(x) = ln(x).

    Domain: x > 0. Derivative: f'(x) = 1/x.

  2. Find the domain and derivative of f(x) = ln(7x).

    Domain: x > 0. Derivative: f'(x) = 7/(7x) = 1/x.

  3. Find the domain and derivative of f(x) = ln(x - 5).

    Domain: x > 5. Derivative: f'(x) = 1/(x - 5).

  4. Find the domain and derivative of f(x) = ln(x^2 + 3).

    Domain: all real numbers (since x^2 + 3 > 0 for all x). Derivative: f'(x) = 2x / (x^2 + 3).

  5. Differentiate f(x) = ln(x^4) using logarithm laws first.

    Rewrite as f(x) = 4*ln(x) for x > 0. Derivative: f'(x) = 4/x.

  6. Find the derivative of f(x) = x * ln(x).

    Domain: x > 0. Use the product rule: f'(x) = (1)*ln(x) + x*(1/x) = ln(x) + 1.

  7. Find the derivative of f(x) = ln((2x + 1)/(x - 3)).

    Rewrite as ln(2x + 1) - ln(x - 3). Derivative: f'(x) = 2/(2x + 1) - 1/(x - 3).