Page 60 - How To Solve Word Problems In Calculus

3
understand what it is, let y = x .
dy dy dx
= ·
dt dx dt
dx
= 3x 2
dt

d dx
3
3
Thus the derivative of x with respect to t, (x ), is 3x 2 .
dt dt
Unless we know the relationship between x and t we cannot
simplify any further.
The result of Example 1 can be stated in a more general
form.

d dx

f (x) = f (x)
dt dt
Of course, the argument of f will not always be x, but the
independent variable is usually t in a related rates problem.

EXAMPLE 2

d 5 4 dw d 1 1 dy d √ 1 dz
(w ) = 5w =− ( z) = √
dt dt dt y y dt dt 2 z dt
2
The following example illustrates the use of the chain rule.
Although not a word problem, it will help clarify basic ideas.

EXAMPLE 3
dx dy
2
3
If y = x + 4x and = 3, compute when x = 1.
dt dt
Solution
dy d 3 2
= (x + 4x )
dt dt
dx
2
= (3x + 8x)
dt
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