Using first principles to find first derivative

[aikastarri]

I'm a 19-year-old uni student from Australia currently studying differential calculus.

I need to find the first derivative of f(x)=x/(x+1)^2 using the first principles definition.

So far all I can do is this:

df/dx=lim h->0 (((x+h)/(x+h+1)^2)-x/(x+1)^2)/h)

When I try to expand this out it becomes so lengthy that I can't keep track of all the numbers. I must be making a mistake somewhere because I don't think my lecturer would be this cruel. My first instinct is to pick out

(x+h+1)^2

as being incorrect because I can't find anything similar anywhere in my notes or online.

So, what exactly am I doing wrong?

 

[Deleted member 4993]

I'm a 19-year-old uni student from Australia currently studying differential calculus.

I need to find the first derivative of f(x)=x/(x+1)^2 using the first principles definition.

So far all I can do is this:

df/dx=lim h->0 (((x+h)/(x+h+1)^2)-x/(x+1)^2)/h)

When I try to expand this out it becomes so lengthy that I can't keep track of all the numbers. I must be making a mistake somewhere because I don't think my lecturer would be this cruel. My first instinct is to pick out

(x+h+1)^2

as being incorrect because I can't find anything similar anywhere in my notes or online.

So, what exactly am I doing wrong?

Your mistake is that - you are trying to find a short-cut instead of paying your dues.

You can break this problem down to two smaller problems - where there will be less-number of terms at a time - but total work is same!!

Breaking into two problems:

\(\displaystyle \displaystyle{f(x) = \frac{x}{(x+1)^2}}\)

\(\displaystyle \displaystyle{f(x) = \frac{1}{(x+1)} - \frac{1}{(x+1)^2}}\)

\(\displaystyle \displaystyle{f(x+h) = \frac{1}{(x+h+1)} - \frac{1}{(x+h+1)^2}}\)

Then continue....

 

[Ishuda]

I'm a 19-year-old uni student from Australia currently studying differential calculus.

I need to find the first derivative of f(x)=x/(x+1)^2 using the first principles definition.

So far all I can do is this:

df/dx=lim h->0 (((x+h)/(x+h+1)^2)-x/(x+1)^2)/h)

When I try to expand this out it becomes so lengthy that I can't keep track of all the numbers. I must be making a mistake somewhere because I don't think my lecturer would be this cruel. My first instinct is to pick out

(x+h+1)^2

as being incorrect because I can't find anything similar anywhere in my notes or online.

So, what exactly am I doing wrong?

Or, just treat x+1 as a function of x, i.e.
\(\displaystyle \frac{df}{dx}\, =\, \underset{h\, \to\, 0}{lim}\,\frac{\frac{x\, +\, h}{(t\, +\, h)^2}\, -\, \frac{x}{t^2}}{h}\)
where t=x+1

EDIT:fix it up, forgot lim as h goes to zero and the division by h. Thnx Subhotosh Khan

 

[Steven G]

I'm a 19-year-old uni student from Australia currently studying differential calculus.

I need to find the first derivative of f(x)=x/(x+1)^2 using the first principles definition.

So far all I can do is this:

df/dx=lim h->0 (((x+h)/(x+h+1)^2)-x/(x+1)^2)/h)

When I try to expand this out it becomes so lengthy that I can't keep track of all the numbers. I must be making a mistake somewhere because I don't think my lecturer would be this cruel. My first instinct is to pick out

(x+h+1)^2

as being incorrect because I can't find anything similar anywhere in my notes or online.

So, what exactly am I doing wrong?

Get common denominators and continue. Remember that algebra (and that is 99.9% of this problem) is a pre-requiste for calculus