Category: Differential Calculus, Algebra
"Published in Vacaville, California, USA"
Find dy/dx and d2y/dx2 by implicit differentiation for:
Solution:
Consider the given equation above
Take the first derivative of the given equation with respect to x by implicit differentiation, we have
Finally, take the second derivative of the above equation with respect to x by implicit differentiation, we have
but
Hence, the above equation becomes
We can further simplify the above equation by using the given equation as follows
Therefore,
Category: Differential Calculus, Algebra
"Published in Vacaville, California, USA"
Given the following functions:
Find dy/dx.
Solution:
The first thing that we need to do is to get the derivative of the given functions with respect to their independent variables.
Take the derivative of the first equation with respect to t, we have
Take the derivative of the second equation with respect to t, we have
Since there are three variables in the given functions, then we have to use the Chain Rule in getting dy/dx, we have
Substitute the values of dy/dt and dx/dt to the above equation, we have
Since the two given functions are simple rational functions, then we can express the above equation in terms of x.
For the given equation,
We can rewrite it in terms of x as follows
Therefore,
Category: Differential Calculus, Algebra
"Published in Vacaville, California, USA"
Given the following functions:
Find dy/dx.
Solution:
The first thing that we need to do is to get the derivative of the given functions with respect to their independent variables.
Take the derivative of the first equation with respect to t, we have
Take the derivative of the second equation with respect to t, we have
Since there are three variables in the given functions, then we have to use the Chain Rule in getting dy/dx, we have
Substitute the values of dy/dt and dx/dt to the above equation, we have
Since the two given equations have higher exponents and it's impossible to express each equations in terms of u, therefore
