Pharmacology for Anaesthesia and Intensive Care

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Section IBasic principles

y

Tangents to the

curve. Values

given by

differentiation

y

AB x

AUC between

times A and B

given by integration

x

(a)

(b)

Figure 6.7.Differentiation and integration.(a) Shows that when the equation for the curve

is differentiated it produces an expression that will give the value for the gradient of the

tangent to the curve at any selected point. (b) When finding the area under the curve (AUC),

it is essential to know the limiting values between which an area is defined. In the example

here, the upper limit is x=B and the lower limit x=A.

Differentiation

Differentiation is a mathematical process used to find an expression that gives the

rate at which the variable represented on the y-axis changes as x changes. When

evaluated, the expression gives the gradient of the tangent to the curve for each

value of x (Figure6.7a). For the function y=f(x) we indicate that differentiation is

needed by writing:

dy/dx.

Because differentiation only tells us about therateat which a function is changing all

functions whose graphs have exactly the same shape, but only differ in their position