__Category__: Differential Calculus, Algebra

"Published in Newark, California, USA"

If y = f(u) and u = ϕ(x), show that
__Solution__:
Consider the given two equations above
Since y is a function of u and u is a function of x, then we can apply the derivative of y with respect to x by Chain Rule Method.
Take the derivative of the first equation with respect to u, we have

Take the derivative of the second equation with respect to x, we have

Hence, by Chain Rule Method,

Take the derivative of the above equation with respect to x, we have

but

Hence, the above equation becomes

Therefore,

where

__Category__: Differential Calculus, Algebra

"Published in Vacaville, California, USA"

Using the fact that
show that
__Solution__:

Consider the given equation above

Take the derivative on both sides of the equation with respect to x by quotient formula, we have

Therefore,

__Category__: Differential Calculus, Algebra

"Published in Vacaville, California, USA"
Find a formula for
__Solution__:
The
first thing that we have to do is to get the derivative of u/v with
respect to x where u and v are functions of x. Apply the derivative by quotient formula, we have

Take the derivative again of the above equation with respect to x by product formula, we have

where: