If y=f(x) and x changes from a to a + [itex]\Delta[/itex]x , we defined the increment of y as
[itex]\Delta[/itex]y = f(a + [itex]\Delta[/itex]x) - f(a)
Accordin to definition of a derivative ,we have
lim [itex]\frac{\Delta y}{\Delta x}[/itex] = f'(a)
so if we denote by [itex]\epsilon[/itex] the difference between Difference Qutient and the derivative we obtain
lim [itex]\epsilon[/itex] = ( [itex] lim \frac{\Delta y}{\Delta x}[/itex] - f'(a) ) = 0
But [itex]\epsilon = \frac{\Delta y}{\Delta x} - f'(a) \Rightarrow \Delta y = f'(a) \Delta x + \epsilon \Delta x[/itex]
If we Define [itex]\epsilon[/itex] to be 0 when [itex]\Delta x[/itex]=0.then [itex]\epsilon[/itex] becomes a continuous function of [itex]\Delta x[/itex]
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.
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.
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.
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.
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.