PHA 5127

4. DIFFERENTIATION

Determination of the rate of change (@ the slope in a graph)

Straight line Curve

m = = constant	Slope not constant, but function of x.
	Instantaneous rate of change for Dx®0
	first derivative of y with respect to x
	dy = differential of y
	dx = differential of x

Rules of differentiation

Examples

y = x²

y =

y = 10x³+ 2x² + 5x + 3

y = ex

y =

5. INTEGRATION

Two applications in pharmacokinetics

1. Determination of area under the curve (AUC)
2. Solution of differential equations
   

Area under the Curve

Determination: sum of narrow slices (rectangles) with the area yd

For

• b = upper limit
• a = lower limit

Rules for integration

Integration is the reverse of differentiation

for n -1

further

Examples

For integration between limits:

• Solve the integral
• Substitute upper and lower limits for the variable and subtract

Example 1 Determine the area under the curve for the following relationship.

y=mx+b

Upper limit = a; Lower limit = 0

Example 2. Determine the area under the curve for the following relationship

Lower limit = 0; Upper limit = t

Rules for Differentiation

y =	dy/dx =
y =	dy/dx =
y = lnx	dy/dx = 1/x

Rules for Integration Between Limits

for:

1. y =

=

2. y =

⁼

3. y =1/x

Trapezoidal Rule

Determination of the AUC from experimental data without integration

The area of a trapezoidal is:

Curve can be thought of as a number of several trapezoidals

Consequently, the area under the curve can be calculated as the sum of its trapezoidals.

Estimation of terminal part of the area by integration:

Example

The second application of integration (solution of differential equations) will be discussed in the next chapter.

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