Differentiation - Calculus

 

 Differentiation

Differentiation is a method of finding the gradient of any point in a polynomial graph. Polynomial graphs are graphs of quadratic, cubic, quartic, etc. equations. By differentiating the polynomial equation, we create another general equation that gives us the gradient of any point on the graph. 




To find the gradient of a specific point on a graph, we usually take the point and draw a tangent from that point. Then we make a right angled triangle of that tangent and use Pythagoras Theorem to find the gradient( the hypotenuse). This method is too long and drawing tangents make it susceptible to human error. This is why we use differentiation! 


How to differentiate:

We do it in parts. We take the power of the value and multiply it to the coefficient. Then we subtract the power by one.

For example:

If y = x4, dy/dx = 4x3


If y = 2x4, dy/dx = 8x3


If y = x5 + 2x-3, dy/dx = 5x4 - 6x-4


dy/dx is the 'symbol' of differentiation.


Question:

In the graph of y=x^4, what is the equation of the tangent of the point where x=7?

1. First we differentiate the equation which gives us 4x^3. 

2. Now we plug in the value of x>>> 4(7^3)

3. We get the gradient as 1372!

Since we know the x- coordinate of the point, we can substitute it in the original equation to find the y coordinate. This will be 7^4=2401! We know the gradient of the tangent (1372) and the coordinates it passes through(7,2401). Now we can even find the equation of the tangent (y=mx+c) at the point where x=7!

The answer should be y= 1372x - 7203


Comments

Post a Comment

Popular posts from this blog

Solving Quadratic Equations ┃Middle Term Splitting

Image
  Middle Term Splitting Apart from plugging in the coefficients in the formula which solves quadratic equations, there are simpler ways of solving quadratic equations, without the use of calculators! The method described here is known as mid-term splitting.  Benefits: This method requires much less calculation and effort. It is solved in a quicker and cleaner method. Drawbacks:  Unlike the quadratic formula, the middle term splitting method only works in certain cases.  How does it work: a x²  + bx + c This method will only work if there are two numbers that add up to 'b' and have a product of 'a*c' What we do is we write the equation in another form and factorise it into two parts which equal zero. The examples below will explain better! Here, 6 and -5 are the two numbers needed. They are the set of numbers which add up to the origibal 'b' whoch was 1, and form a product of -30( a*c)! After that the equation was just factorised. Since this would equal to zero, ...
Read more

The Fibonacci Sequence

Image
  The Fibonacci When one of  "the most talented Western mathematicians of the Middle Ages", Leonardo Bonnaci, created the Fibonacci sequence, it became widely popular and since then, it has held its value as one of the  most famous mathematical concepts to ever exist amongst learners. It has impacted the world immensely, its influence ranging from finance and coding theories to even the shape of a simple flower! The Sequence: In the Fibonacci, each term is made up of the sum of its previous two terms.  N= (N-1) + (N-2) where 'N' is the term number. The  Fibonacci  sequence goes as follows: 0,1,1,2,3,5,8,13,21,34,55..... For example, lets take the 7th term which is 8. This implies that in our case N=7. The 2 terms before it, the 5th and 6th terms are 3 and 5  respectively , which add up to 8.  The next term would be the sum of 34 and 55, so the 12th term is 89!  The sequence goes on till   infinity . Applications in real life: T...
Read more

Finding the Areas of shapes in graphs

Image
  In a graph, there are numerous points which join to form various polygons. These polygons may be irregular in shape, with any number of sides. The area of these polygons can be found using different methods! If we have a regular polygon, we can use that polygon's formula to find the area. So if we have a hexagon, we can just use the formula :                                              a^2(3√3)/2 This is the formula to find the area of a regular hexagon. 'a' represents the length of a side. There is a different formula for different regular polygons.  The side length of a polygon can be found using the distance formula, where we calculate the distance between two points. But, what if the polygons are not regular? If they have different side lengths and different angles on each side? The formula generally used here is commonly known as the sh...
Read more