Derivative Differentiation

what is differentation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean constant rate change ?

the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. And the slope (gradient) is always 300/5 = 60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

What does this mean when rate of change is not constant ?

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive. Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it become 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

derivative differentiation -The Derivative

The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be approached in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change). Historically there was (and maybe still is) a fight between mathematicians which of the two illustrates the concept of the derivative best and which one is more useful. We will not dwell on this and will introduce both concepts. Our emphasis will be on the use of the derivative as a tool.

the physical concept of derivatives

This approach was used by Newton in the development of his Classical Mechanics. The main idea is the concept of velocity and speed. Indeed, assume you are traveling from point A to point B, what is the average velocity during the trip? It is given by

Average velocity = distance from A to B / time to get from A to B.

If we now assume that A and B are very close to each other, we get close to what is called the instantaneous velocity. Of course, if A and B are close to each other, then the time it takes to travel from A to B will also be small. Indeed, assume that at time t=a, we are at A. If the time elapsed to get to B is $\Delta t$, then we will be at B at time $t=a + \Delta t$. If $\Delta s$ is the distance from A to B, then the average velocity is

\begin{displaymath}\mbox{Average velocity} = \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

The instantaneous velocity (at A) will be found when $\Delta t$get smaller and smaller. Here we naturally run into the concept of limit. Indeed, we have
\begin{displaymath}\mbox{Instantaneous Velocity (at A)} = \lim_{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\cdot\end{displaymath}

derivative differentiation - formulas

General Derivative Formulas:
1)  Where  is any constant.
2)  It is called Power Rule of Derivative.
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4)  Power Rule for Function.
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9)  It is called Product Rule.
10)  It is called Quotient Rule.

Derivative of Logarithm Functions:

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Derivative of Exponential Functions:

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Derivative of Trigonometric Functions:

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Derivative of Hyperbolic Functions:

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Derivative of Inverse Trigonometric Functions:

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Derivative of Inverse Hyperbolic Functions:

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