# How is Epsilon Defined in the Proof of the Chain Rule in James Stewart Calculus?

• mahmoud2011
In summary, the conversation discusses the definition of a derivative and the introduction of a property of differentiable functions. It also addresses the issue of defining epsilon to be 0 when delta x = 0 and its continuity as a function of delta x. There is a comparison made to a piecewise function.
mahmoud2011
From james stewart calculus Early Transcendentals.Before he states the proof he intoduced a property of differentiable funcion
If y=f(x) and x changes from a to a + $\Delta$x , we defined the increment of y as

$\Delta$y = f(a + $\Delta$x) - f(a)

Accordin to definition of a derivative ,we have

lim $\frac{\Delta y}{\Delta x}$ = f'(a)​

so if we denote by $\epsilon$ the difference between Difference Qutient and the derivative we obtain

lim $\epsilon$ = ( $lim \frac{\Delta y}{\Delta x}$ - f'(a) ) = 0

But $\epsilon = \frac{\Delta y}{\Delta x} - f'(a) \Rightarrow \Delta y = f'(a) \Delta x + \epsilon \Delta x$

If we Define $\epsilon$ to be 0 when $\Delta x$=0.then $\epsilon$ becomes a continuous function of $\Delta x$

My problem is how we defined $\epsilon$ to be 0 when $\Delta x$=0

where this is not in the Domain.

what is the problem?

Like a piece wise Functions

## 1. What is the chain rule?

The chain rule is a mathematical rule used in calculus to find the derivative of a composite function. It helps to break down complex functions into smaller, simpler functions that are easier to differentiate.

## 2. Why is the chain rule important?

The chain rule is important because it allows us to find the derivative of composite functions, which are commonly used in real-world applications. It is also a fundamental concept in calculus and is necessary for solving more advanced problems.

## 3. How do I apply the chain rule?

To apply the chain rule, you first need to identify the inner and outer functions of a composite function. Then, take the derivative of the outer function and multiply it by the derivative of the inner function. This will give you the derivative of the composite function.

## 4. Can you give an example of using the chain rule?

Sure, let's say we have the function f(x) = (2x + 5)^3. The inner function is 2x + 5 and the outer function is x^3. To find the derivative, we use the chain rule: f'(x) = 3(2x + 5)^2 * 2 = 6(2x + 5)^2.

## 5. How can I remember the chain rule?

One mnemonic device to remember the chain rule is "outer derivative, inner times derivative of the inner". Another helpful tip is to always start with the outer function and work your way inwards when applying the chain rule.

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