Question

Use the definition of the derivative to show f^ prime (x)=2x+2 iff(x)=x^ 2 +2x This long way" by substituting into the limit: f^ prime (x)=lim h 0 f(x+h)-f(x) h

Answer 1

f(x) = x^2 +2

We need to use the definition of derivative

The first step is to find f(x+h)

f(x+h) = ( x+h)^2 + 2(x+h)

FOILing and distributing the second term

= x^2 + 2xh + h^2 + 2x + 2h

Now we subtract f(x) from this

f(x+h) - f(x) = x^2 + 2xh + h^2 + 2x + 2h - ( x^2 + 2x)

Distribute the minus sign and subtract like terms

= x^2 + 2xh + h^2 + 2x + 2h - x^2 - 2x

= 2xh + h^2 + 2h

Now this term is the numerator of the definition and we divide by h

As h goes to 0 the second term goes to zero and we are left with

2x+2