Answer to Question #142795 in Calculus for liam donohue

Question

f(x+h)−f(x)=7hx2−5hx+1h2x+4h2+7h3.
Find f′(x).

Answer 1

$\displaystyle f(x+h)−f(x) =7hx^2−5hx+h^2x+4h^2+7h^3 \\ \frac{f(x+h)−f(x)}{h} =7x^2−5x+hx+4h +7h^2\\ \begin{aligned} \lim_{h \rightarrow 0} \frac{f(x+h)−f(x)}{h} &= \lim_{h \rightarrow 0}(7x^2−5x+hx+4h +7h^2) \\&= 7x^2 - 5x + 0 + 0 + 0 = 7x^2 - 5x \end{aligned}\\ \therefore f'(x) = 7x^2 - 5x$