Definition of Differentiation

Introduction

Its morning. You get into your car and you drive to your work. Its 50km away and you need 1 hour to get there. So your average speed is u=s/t=50Km/hour. But when you drive in the city you drive slowly and when you drive in the highway you drive quickly. In order to calculate your speed at any point, the “instant” speed, you need to divide a small part of the trip that you are interested in calculating by the time you need to travel over it. So if you travel 1Km in 0.01 hours (36 seconds) your local speed will be u=1/0.01=100Km/hour. The smaller the interval you are calculating, the more accurate your calculation will be.

In a more formal way, consider the function y=f(x).

Partial Derivation

Introduction

In many cases a function depends on more than one independent variables. In order to calculate the derivative of this function, with respect to one variable, we make the calculations as usual, considering all the other variables as constants.

Separable Variables Differential Equations

Separable variables differential equations of first order have the following form:

As we can see, the independent variable x is totally separate from the unknown function y.

In order to solve this kind of differential equations, we integrate both sides of the equation:

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