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In calculus, the derivative represents the rate at which a function's value changes as its input changes. This concept is fundamental, as it provides a way to understand how a function behaves locally—that is, at or near a specific point on its graph.

### Calculating the DerivativeThe derivative of a function \( f \) at a point \( x \) is expressed as \( f'(x) \). It is found using the limit definition:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)

To compute \( f'(x) \), substitute the difference quotient in the expression and evaluate the limit as \( h \) approaches zero.

### ExampleTo solidify this understanding, apply the derivative definition to the function \( f(x) = x^2 \):

• \( f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} \)
• This simplifies to \( 2x \) when \( h = 0 \).

This result implies that for every small increase in \( x \), the function's value, \( x^2 \), increases approximately by \( 2x \).

Understanding derivatives is crucial for analyzing and predicting patterns of change in various fields, such as physics and economics.

A limit in calculus is used to define the value that a function \( f(x) \) approaches as the input \( x \) approaches some value. It forms the foundation of differentiability and the calculation of derivatives.

### Understanding LimitsA limit describes the behavior of a function; it doesn't necessarily tell you the value of the function at the point. The formal notation used is:

• \( \lim_{x \to c} f(x) = L \)

This expression means that as \( x \) gets closer to \( c \), \( f(x) \) gets closer to \( L \). Limits help deal with points of discontinuity, where a function might not exist at \( c \) but still approaches a value.

### ExampleConsider the function \( f(x) = \frac{x^2 - 1}{x - 1} \).

• Substituting \( x = 1 \) gives a division by zero, undefined directly.
• However, by factoring, \( f(x) = x + 1 \) for all \( x eq 1 \).
• The limit as \( x \to 1 \) is \( \lim_{x \to 1} f(x) = 2 \).

The limit exists even though the function isn't defined at \( x = 1 \), demonstrating how limits provide insights into function behavior beyond simple values.

Limits are essential for understanding continuous changes and are instrumental in defining concepts such as derivatives and integrals.

Function differentiability is an extension of understanding limits and derivatives. A function is said to be differentiable at a point if its derivative exists at that point.

### Requirements for DifferentiabilityDifferentiability implies that:

• The function is continuous at that point.
• The limit of its derivative, derived from the difference quotient, exists at that point.

If a function is differentiable at every point in its domain, it's called a differentiable function.

### ExampleTake the simple function \( f(x) = x^3 \):

• It's differentiable at all points because \( f'(x) = 3x^2 \) is defined everywhere.

### Differentiability vs. ContinuityWhile continuity is necessary for differentiability, not all continuous functions are differentiable. For instance, the function \( f(x) = |x| \) is continuous everywhere, but not differentiable at \( x = 0 \) due to a sharp corner there.

Understanding differentiability is crucial as it allows for smooth approximations of functions, which are vital in real-world applications like machine learning, physics simulations, and more.