I do not understand this:
Evaluate and simplify f(x+h)-f(x)/h if
f(x)=x^2-2x
f(x+h) = (x+h)^2 - 2(x+h)
= x^2 + 2hx + h^2 - 2x - 2h
so f(x+h)-f(x)/h
= (x^2 + 2hx + h^2 - 2x - 2h - (x^2-2x))/h
= (2hx + h^2 - 2h)/h
= 2x + h - 2
tutorial(dot)math(dot)lamar(dot)edu/Classes/CalcI/DefnOfDerivative.aspx
lim h->0 ([(x+h)^2-2(x+h)]-[x^2-2x])/h
lim h->0 (2x + h - 2)
To evaluate and simplify the expression f(x+h) - f(x)/h, we need to substitute the given expression for f(x) into the formula and simplify it.
Step 1: Substitute the expression for f(x) into the formula:
f(x) = x^2 - 2x
f(x + h) = (x + h)^2 - 2(x + h)
Step 2: Simplify the expression inside the parentheses:
f(x + h) = (x^2 + 2xh + h^2) - 2(x + h)
Step 3: Distribute the negative sign across the parentheses:
f(x + h) = x^2 + 2xh + h^2 - 2x - 2h
Step 4: Expand and combine like terms:
f(x + h) = x^2 - 2x + 2xh - 2h + h^2
Step 5: Substitute the expressions back into the original formula:
[f(x + h) - f(x)] / h = [(x^2 - 2x + 2xh - 2h + h^2) - (x^2 - 2x)] / h
Step 6: Simplify the numerators:
[f(x + h) - f(x)] / h = (2xh - 2h + h^2) / h
Step 7: Cancel out the common factor of h in the numerator and denominator:
[f(x + h) - f(x)] / h = 2x - 2 + h
Thus, the simplified expression for f(x + h) - f(x)/h is 2x - 2 + h.