find the derivative of f(x)=x^2 + x -3 using the definition of the derivative

Question

callisto · Answer

$$f'(x)$$$$=\lim_{h \rightarrow 0}\frac{f(x+h) - f(x)}{h}$$$$=\lim_{h \rightarrow 0}\frac{[(x+h)^2 + (x+h) -3] -(x^2 + x -3)}{h}$$Can you simplify the fraction first?

anonymous · Answer

I used x^2 + 2(x)(h) + h^2 + x +h -3 -x^2 -x +3 = 2x + x ??

callisto · Answer

x^2 + 2(x)(h) + h^2 + x +h -3 -x^2 -x +3 <- correct for denominator.
But it's not equal to 2x + x...

$$x^2 + 2(x)(h) + h^2 + x +h -3 -x^2 -x +3$$Group the like terms together:
$$x^2  -x^2 + x  -x -3  +3+ h^2+ 2(x)(h)+h$$
Now, can you simplify the above expression

anonymous · Answer

basically factor out the h and H(h+2x+1) divided by the h on denominator and = h + 2x + 1?

callisto · Answer

Yes.

callisto · Answer

And you got:
$$\lim_{h \rightarrow 0} (h+2x+1)$$Can you evaluate the limit?

anonymous · Answer

Would be 2x + 1 ??? Because h is going to 0

callisto · Answer

Indeed it is!

anonymous · Answer

Holy Cow! It came together finally!! Thank you very much for your help! I would have been stuck!!!

callisto · Answer

You're welcome :)

anonymous · Answer

Have a great day!!

callisto · Answer

You too! :)