Find Derivative of 3/x+2 using f(x+h) - (fx) / h

[pooker]

Our teacher wants us to find derivatives the long way

f(x+h) - (fx) / h

So anyways, on one of my questions 3 / x + 2 , x=8

I know how to find the derivative of 3 / x+2 , but why was their an x=8 next to it? Am I supposed to plug this in after finding the derivative?

 

[mathstudent88]

yes, you need to plug it in for x after you find the derivative.

 

[HallsofIvy]

I disagree with mathstudent88. The basic definition of 'derivative' is 'derivative at a given point'. From what you say, I imagine your teacher is expecting you to put it directly into the definition of the derivative of f at x₀:
\lim_{h\rightarrow 0} \frac{f(x_0+ h)- f(x_0)}{h}
which, for f(x)= 3/(x+2), at x₀= 8 is
\lim_{h\rightarrow 0}\frac{3/((8+h)+2)- 3/(8+2)}{h}
= \lim_{h\rightarrow 0}\frac{3/(10+h)- 3/10}{h}

 

[Office_Shredder]

With the obvious substitution of x=8 instead of x=2

 

[HallsofIvy]

Thanks, OfficeShredder, I've edited that mistake. (So, now I can pretend I never made it!)