All things calculus

Introduction :

First principles have helped us derive the formula for differentiation which has numerous applications. The first derivative gives us the gradient function and the second let's us know the nature of a certain point in our function although we just repeat the process.

If an x-value is inserted into the first derivative, we can obtain the gradient at that point of the function. If it is inserted into the second derivative, a negative return means it is a maximum point and a positive return means it is a minimum point.

Calculating turning or stationary points :

At times, we want to know the maximum or minimum of a scenario like the least time taken for an event to occur or the highest amount of interest we can get. To know this point, we need to let the derivative to equal zero as that is the point when the gradient equals zero.

Calculating rates of change :

This invention can also help us determine how something like area or volume can change with time. Using chain rule, we are able to connect rates with each other and determine the maximum or minimum of a scenario.

For example, if we want to know how fast a pierced bottle is leaking, we just have to use the formula :

dy/dx = dy/dt * dt/dx which shows the chain rule clearly if one is familiar.

Reversing differentiation (Integration)

Reversing the process is easy can one can remember by differentiating a similar function to compare. This allows us to get the equation from the gradient function but with an unknown constant, normally represented by c. We can determine the unknown constant by using any known coordinate of the graph namely, the x and y values.