Show that applying the chain rule to z = x / y , where x and y are arbitrary functions of t...

Question:

Show that applying the chain rule to {eq}z= \frac{x}{y} {/eq}, where {eq}x {/eq} and {eq}y {/eq} are arbitrary functions of {eq}t {/eq}, given the quotient rule for functions of one variable.

Quotient Rule:

The quotient rule is one of the general rules that allow you to calculate the derivative of complex expressions.

That is, from the following general formula:

{eq}z\left( t \right) = \frac{{x\left( t \right)}}{{y\left( t \right)}},\quad z'\left( t \right) = \frac{{x'\left( t \right)y\left( t \right) - x\left( t \right)y'\left( t \right)}}{{{y^2}\left( t \right)}} {/eq}.

Answer and Explanation:

Written the function as a function of the variable {eq}t {/eq}, {eq}z = \frac{x}{y} \to z\left( t \right) = \frac{{x\left( t \right)}}{{y\left( t \right)}} {/eq}.

Applying the quotient rule, we have:

{eq}z'\left( t \right) = \frac{{x'\left( t \right)y\left( t \right) - x\left( t \right)y'\left( t \right)}}{{{y^2}\left( t \right)}} {/eq}.