Finding The Equation Of A Tangent Line & Calculating A Derivative using the defenition

of A Derivative

                        by: Lazaro Reyes

Calculating a derivative using the defenition of a derivitive

Before finding the equation of a tangent line we must first understand Derivative. I will show and explain problems for better understanding. The definition that i find better to work with is the following.      It basically says that f prime of x = the limit as h goes to 0, f of x+h- f of x over h.

Let us find the derivative of the quantity .........f(x)=12+7x

• Every where theres x replace it with x+h. so the first part looks like this 12+7(x+h)

• Then you subtract it by the original quantity (12+7x)

• In this case h will equal 0• Then you will multiply it out. so

it becomes 12+7x+7h-7x over h.

• Then you cancel all the like terms and you are left with 7h over h

• Cancel h to leave 7 alone and 7 becomes your answer.

  

Lets try some more problems........G(t)=4t over t+1

• Using the definition of derivative says that we must replace t for (t+h).

• After you have done that your equation will look like this 4(t+h) over t+h+1.

• Then you subtracted by its original equation and divide by h.

• Then you look for the common denomanator to simplify.so the equation will now look like this ...

• After doing that you will cancel all the like terms. So you will end up with 4h over (t+h+1) (t+1).

• Get rid of h you will have to change h to h over 1 and then put one on top of the h so it can allow it to cacel it out.

• After doing this you will end up with G prime of (t) = 4 over (t+1) (t+1) which can be equal to 4 over (t+1)squared.

Finding the equations of Tangent Lines

• One thing that is really necessary is to know the point-slope formula which is y-y1=m(x-x1).

• The other thing is derivative.

• In the equations i will demonstrate, you will see the steps required to obtain the equation of the Tangent Lines. 

 

 

Finding the Equation of a Tangent Line at points (2,4) on y=x squared

• The point-slope formula is needed for these type of problems. ........... y - y1= m (x-x1)

• You have your two points (2,4) so keep that in mind.

• Now, the derivative of x squared is 2x and we are getting this from y= x squared.

• Since we already know our x point or coordinate.we plug it in for every x.

• After doing this you will get 4 and that is your slope.

• Now we take our point slope formula and we plug in what we already know.

• Soon your equation will look like this :         y - 4=4 (x-2)

• Then you solve it first on the right side.• you will get y-4=4x-8 so you add the for from

the left to the right leaving y alone• when you solve it piece by piece you will soon

get your answer y=4x-4 

The problem continue...• Again, your points are (1,9) and this will

be on f (x)=(1+2x)squared.• something to keep in mind is that the

derivative of 1 is 0 and de derivative of positive 2x is positive 2.

• To start of we will set the equation to f prime of x =2(1+2x)to the power of 1 (2).

• because you take the power and put it in front and the inside you leave it all to the power of 1 and we saide the derivative of 2x is 2 so we put that after.

• Then you multiply two by the outside 2 which gives you 4 (1+2x)

• Then we plug in what we know x = 1• Then you solve thee inside first which will

give us 3• Then multiply and we get 12 • so y - 9=12 (x-1) and we could put it in

standard form if asked...

Thank You for listening...