Related rates

In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Differentiation with respect to time or one of the other variables requires application of the chain rule, since most problems involve several variables.
Fundamentally, if a function



F


{\displaystyle F}
is defined such that



F
=
f
(
x
)


{\displaystyle F=f(x)}
, then the derivative of the function



F


{\displaystyle F}
can be taken with respect to another variable. We assume



x


{\displaystyle x}
is a function of



t


{\displaystyle t}
, i.e.



x
=
g
(
t
)


{\displaystyle x=g(t)}
. Then



F
=
f
(
g
(
t
)
)


{\displaystyle F=f(g(t))}
, so





F
′

=

f
′

(
g
(
t
)
)
⋅

g
′

(
t
)


{\displaystyle F'=f'(g(t))\cdot g'(t)}
Written in Leibniz notation, this is:







d
F


d
t



=



d
f


d
x



⋅



d
x


d
t



.


{\displaystyle {\frac {dF}{dt}}={\frac {df}{dx}}\cdot {\frac {dx}{dt}}.}
Thus, if it is known how



x


{\displaystyle x}
changes with respect to



t


{\displaystyle t}
, then we can determine how



F


{\displaystyle F}
changes with respect to



t


{\displaystyle t}
and vice versa. We can extend this application of the chain rule with the sum, difference, product and quotient rules of calculus, etc.
For example, if



F
(
x
)
=
G
(
y
)
+
H
(
z
)


{\displaystyle F(x)=G(y)+H(z)}
then







d
F


d
x



⋅



d
x


d
t



=



d
G


d
y



⋅



d
y


d
t



+



d
H


d
z



⋅



d
z


d
t



.


{\displaystyle {\frac {dF}{dx}}\cdot {\frac {dx}{dt}}={\frac {dG}{dy}}\cdot {\frac {dy}{dt}}+{\frac {dH}{dz}}\cdot {\frac {dz}{dt}}.}

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