### Related rates

In this section, most functions will be functions of a parameter $t$
which we will think of as *time*. There is a convention coming
from physics to write the derivative of any function $y$
of $t$ as $\dot{y}=dy/dt$, that is, with just a *dot* over the
functions, rather than a *prime*.

The issues here are variants and continuations of the previous
section's idea about *implicit differentiation*. Traditionally,
there are other (non-calculus!) issues introduced at this point,
involving both story-problem stuff as well as requirement to be able
to deal with *similar triangles*, the *Pythagorean Theorem*,
and to recall formulas for *volumes* of cones and such.

Continuing with the idea of describing a function by a relation, we
could have *two* unknown functions $x$ and $y$ of $t$, *related* by some formula such as
$$x^2+y^2=25$$
A typical question of this genre is ‘What is $\dot{y}$ when $x=4$ and
$\dot{x}=6$?’

The fundamental rule of thumb in this kind of situation is *differentiate the relation with respect to $t$*: so we differentiate
the relation $x^2+y^2=25$ with respect to $t$, even though we don't
know any details about those two function $x$ and $y$ of $t$:
$$2x\dot{x}+2y\dot{y}=0$$
using the chain rule. We can solve this for $\dot{y}$:
$$\dot{y}=-{x\dot{x} \over y}$$
So *at any particular moment*, if we knew the values of
$x,\dot{x},y$ then we could find $\dot{y}$ *at that moment*.

Here it's easy to solve the original relation to find $y$ when $x=4$: we get $y=\pm 3$. Substituting, we get $$\dot{y}=-{4\cdot 6\over \pm 3}=\pm 8$$ (The $\pm$ notation means that we take $+$ chosen if we take $y=-3$ and $-$ if we take $y=+3$).

#### Exercises

- Suppose that $x,y$ are both functions of $t$, and that $x^2+y^2=25$. Express ${ dx \over dt }$ in terms of $x,y,$ and ${ dy \over dt }$. When $x=3$ and $y=4$ and ${ dy \over dt }=6$, what is${ dx \over dt }$?
- A 2-foot tall dog is walking away from a streetlight which is on a 10-foot pole. At a certain moment, the tip of the dog's shadow is moving away from the streetlight at 5 feet per second. How fast is the dog walking at that moment?
- A ladder $13$ feet long leans against a house, but is sliding down. How fast is the top of the ladder moving at a moment when the base of the ladder is $12$ feet from the house and moving outward at $10$ feet per second?

#### Thread navigation

##### Calculus Refresher

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