Hi, I need help with this problem, please!

Question

Find the derivative of f(x)=x^2+x using the limit process. I know the answer would be 2x+1, but I don't know how to prove it using the limit process. Thank you!

Answer 1

The way to do this is:

lim h->0 [f(x+h)-f(x)]/h

Using the function you provided:

[(x+h)²+(x+h)-(x²+x)]/h FOIL the first term and distribute the -1

[x²+2xh+h²+x+h-x²-x]/h Simplify the numerator by combining like terms

[2xh+h²+h]/h Factor an h in the numerator

h(2x+h+1)/h Cancel the h in the numerator and denominator

2x+h+1

Now, I will evaluate using the limit

The way to do this is:

lim h->0 2x+h+1

2x+1