**The Derivative using Increment Method**

The increment method is a technique used in
calculus to derive mathematical formulas that describe the behavior of a
function as its input variable changes. The technique is particularly useful in
cases where the function is not easily differentiable or when the function is
defined only in terms of discrete values.

The basic idea behind the increment method is to calculate the
difference between two values of a function as the input variable changes by a
small amount, which is referred to as the increment. By taking the limit of
this difference as the increment approaches zero, we can obtain the derivative
of the function at a particular point.

The advantage of using the increment method is that it allows us
to approximate the behavior of a function using only the information that is
readily available. For example, if we know the values of a function at discrete
points, we can use the increment method to derive a formula that describes the
rate at which the function changes between those points.

Another advantage of the increment method is that it can be used to derive formulas for functions that are not continuous or differentiable at certain points. For example, if a function has a jump discontinuity at a particular point, we can use the increment method to derive a formula that describes the behavior of the function on either side of the point.

**Illustrative Examples**

**Find the first derivative using the increment method.**

**1.)**

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*cancel all dark greens*

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*cross out*

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*answer*

*2.)*

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*remember y as given eq,n?*

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*compensate similar terms*

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*answer*

*3.)*

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*this y is the given eq'n.*

*, substituting the value of y*

*compensate dark green color*

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*cross out*

,

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*answer*

**Example problems with solutions:**

**Find the first derivative using the increment method.**

*These problems can be found in the textbook "Differential and Integral Calculus by Love and Rainville"*

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*y from the given eq'n*

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*substitute the value of*

*y*

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*removing parenthesis and collecting/compensating for similar terms.*

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*compensate all in red*

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*cross out*

,

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*answer*

,

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*answer*

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*answer*

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*, answer*

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*, answer***Seatwork with Partial Solutions:**

**Find the first derivative using the increment method.**

__Click to continue__

__Click to continue__

__Click to continue__

**Simple Quiz:**

**Find the first derivative using the increment method.**

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Second derivatives

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## + comments + 8 comments

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