If the problem doesn't specify the domain of definition and asks you to find the derivative of a function, the usual approach is:

1. Assume the function is defined wherever it is mathematically valid.

   • If the function has restrictions (e.g., a denominator that cannot be zero, or a square root that requires non-negative values), consider these restrictions when differentiating.
   • However, explicitly stating any assumptions about the domain is good practice.

2. Differentiate the function as if it is defined in its natural domain.

   • Apply the standard rules of differentiation, such as the product rule, quotient rule, chain rule, or power rule.

3. State any conditions for validity.

   • After differentiation, clarify where the derivative is valid, based on the domain of the original function.

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Example:

Function: $f(x) = \frac{1}{x}$

1. The natural domain of $f(x)$ is $x \neq 0$ (denominator cannot be zero).
2. Derivative: $f'(x) = -\frac{1}{x^2}$
3. State the domain of the derivative: $x \neq 0$.

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If the function has no explicit restrictions and is valid everywhere (e.g., $f(x) = x^2$), differentiate normally.

Would you like more details or examples? Here are related questions to consider:

1. What should I do if the derivative involves a piecewise function?
2. How do I handle derivatives for functions with implicit domains, like trigonometric functions?
3. What happens when the function involves absolute values?
4. How do I include domain considerations in multi-variable calculus?
5. What are the steps to find the derivative for functions involving logarithms or exponentials?

Tip: Always double-check for any implied domain restrictions, such as denominators or square roots!