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The domain of a function consists of all possible input values (usually represented by \( x \)) for which the function is defined. For example, consider the function \( g(x) = \sqrt{1 + 2x} \). Here, the square root function \( \sqrt{u} \) is only defined for non-negative values. Thus, we ensure the expression inside the square root is greater than or equal to zero.

We solve the inequality \( 1 + 2x \geq 0 \). By subtracting 1 from both sides, we get \( 2x \geq -1 \). Dividing both sides by 2 yields \( x \geq -0.5 \).

This means the domain of \( g(x) \) is all real numbers \( x \) for which \( x \geq -0.5 \), or in interval notation, \([-0.5, \infty)\).

• The domain tells us valid input values
• Even basic functions like sqrts can limit domain

The domain of a derivative may differ from that of the original function, depending on the behavior of the derivative itself.

Rationalizing the numerator is a technique used to simplify expressions that involve square roots, especially when they appear in a fraction's numerator.

Consider our derivative expression for \( g(x) = \sqrt{1+2x} \):\[ \frac{\sqrt{1 + 2(x+h)} - \sqrt{1 + 2x}}{h} \].

This can be tricky to evaluate directly, so we multiply the numerator and the denominator by the conjugate \( \sqrt{1 + 2(x+h)} + \sqrt{1 + 2x} \).

This results in:

\[ \frac{(\sqrt{1 + 2x + 2h} - \sqrt{1 + 2x})(\sqrt{1 + 2x + 2h} + \sqrt{1 + 2x})}{h(\sqrt{1 + 2x + 2h} + \sqrt{1 + 2x})} \]

By applying the difference of squares (\( a^2 - b^2 = (a-b)(a+b) \)), the expression simplifies the numerator to \( 2h \).

Canceling \( h \) in the numerator and denominator simplifies things considerably:

\[ \frac{2}{\sqrt{1 + 2x + 2h} + \sqrt{1 + 2x}} \]

• Rationalization is a powerful tool
• Used to handle limits with radicals

This simplification is often crucial for finding limits and derivatives efficiently.

Limit calculation is the cornerstone of differential calculus, especially when defining derivatives. The derivative of a function at a point \( x \), \( g'(x) \), is defined using the difference quotient:

\[ \lim_{h \to 0} \frac{g(x + h) - g(x)}{h} \]

For \( g(x) = \sqrt{1 + 2x} \), this becomes:

\[ \lim_{h \to 0} \frac{\sqrt{1 + 2x + 2h} - \sqrt{1 + 2x}}{h} \].

After rationalizing the numerator and simplifying, we arrive at:

\[ \lim_{h \to 0} \frac{2}{\sqrt{1 + 2x + 2h} + \sqrt{1 + 2x}} \]

When \( h \to 0 \), the term \( \sqrt{1 + 2x + 2h} \) tends to \( \sqrt{1 + 2x} \). Substituting, we get:

\[ \frac{2}{2\sqrt{1 + 2x}} = \frac{1}{\sqrt{1 + 2x}} \].

This value is the derivative of \( g(x) \) at any given \( x \) in its domain.

• Limits find behavior right as variables approach a point
• Essential for calculating exact derivatives

Understanding limits is crucial for mastering derivatives and their applications.