91Ó°ÊÓ

The concept of the derivative is central to calculus. It's a way to measure how a function changes as its input changes. For a given function, the derivative at a certain point tells us the slope of the tangent line to the curve at that point.
In mathematical terms, the derivative of a function \( f(x) \) at a point \( a \) is defined as:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
This definition is known as the limit definition of the derivative. It uses the concept of a limit to describe the rate of change of the function as the input changes by an infinitesimally small amount \( h \).
When working with the function \( f(x) = \sqrt{x} \), we start by plugging it into this definition of the derivative.

The limit process is a fundamental step in finding derivatives using the definition. After substituting \( f(x) = \sqrt{x} \) into the derivative definition, we get:
\( f'(a) = \lim_{h \to 0} \frac{\sqrt{a + h} - \sqrt{a}}{h} \)
Here, we need to evaluate what happens to the expression as \( h \) gets very close to 0. This can be tricky because the numerator \( \sqrt{a + h} - \sqrt{a} \) and the denominator \( h \) both tend to 0, leading to an indeterminate form \( \frac{0}{0} \). To resolve this, further algebraic manipulation is required, such as rationalizing the numerator.

Rationalizing the numerator is a useful algebraic technique to simplify expressions involving square roots. In our case, we multiply the numerator and the denominator by the conjugate of the numerator. The conjugate of \( \sqrt{a + h} - \sqrt{a} \) is \( \sqrt{a + h} + \sqrt{a} \). Multiplying both the numerator and the denominator by this conjugate gives:
\( f'(a) = \lim_{h \to 0} \frac{(\sqrt{a + h} - \sqrt{a})(\sqrt{a + h} + \sqrt{a})}{h(\sqrt{a + h} + \sqrt{a})} \)
This manipulation helps us eliminate the square roots in the numerator by turning it into a difference of squares.

The difference of squares is a special algebraic formula:
\( (x - y)(x + y) = x^2 - y^2 \)
In our context, applying this formula to \( (\sqrt{a+h} - \sqrt{a})(\sqrt{a+h} + \sqrt{a}) \) gives:

\( (\sqrt{a+h})^2 - (\sqrt{a})^2 \)
Which simplifies to \( a + h - a = h \).
So, our expression now looks like:
\( f'(a) = \lim_{h \to 0} \frac{h}{h(\sqrt{a + h} + \sqrt{a})} \)
We're now in a position to cancel out any common factors.

Once we simplify the expression and find common factors, we progress to evaluate the limit. Cancelling out \( h \) in the numerator and denominator gives:
\( f'(a) = \lim_{h \to 0} \frac{1}{\sqrt{a + h} + \sqrt{a}} \)
As \( h \) approaches 0, \( \sqrt{a + h} \) approaches \( \sqrt{a} \). So the limit becomes:

\( f'(a) = \frac{1}{\sqrt{a} + \sqrt{a}} = \frac{1}{2\sqrt{a}} \)
Therefore, the derivative of the square root function is:
\( f'(a) = \frac{1}{2\sqrt{a}} \)