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Home Differentiation Differentials and Tangent Lines Examples Example 3
 
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Example 3

Whenever a derivative f'(x) is known, we can find the differential dy at once by simply multiplying the derivative by dx, using the formula dy = f'(x) dx. The examples in the last section give the following differentials.

(a)

y = x3

dy = 3x2 dx.

(b)

y = √x,

02_differentiation-57.gif where x > 0.

(c)

y = 1/x

dy = -dx/x2 when x ≠ 0

(d)

y = |x|

dy = dx when x > 0,
dy = -dx when x < 0
dy = undefined when x = 0

(e)

y = bt - 16t2

dy = (b - 32t)dt

The differential notation may also be used when we are given a system of formulas in which two or more dependent variables depend on an independent variable. For example if y and z are functions of x,

y = f (x), z = g(x),

then

Δy, Δz, dy, dz are determined by

Δy = f (x + Δx) - f(x),

Δz = g(x + Δx) - g(x),

dy = f'(x) dx,

dz = g'(x) dx.
Home Differentiation Differentials and Tangent Lines Examples Example 3

Last Update: 2010-11-25