Math Insight

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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 $y=+3$).


  1. 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${ dx \over dt }$?
  2. 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?
  3. 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?