Question

Given the function f(x)=-x^2+3x, express the value of {f(x+h)-f(x)}{h}​ in simplest form.

Answer 1

Answer:

$f'(x) = 3-2\cdot x$

Step-by-step explanation:

Let be $f(x) = -x^{2}+3\cdot x$, such that:

$f'(x) = \frac{f(x+h)-f(x)}{h}$

Hence,

$f(x+h) = -(x+h)^{2}+3\cdot (x+h)$

$f(x+h) = -(x^{2}+2\cdot x\cdot h + h^{2})+3\cdot x + 3\cdot h$

$f(x+h) = -x^{2}+(3-2\cdot x) \cdot h - h^{2}+3\cdot x$

$f(x) = -x^{2}+3\cdot x$

Then,

$f'(x) = \frac{-x^{2}+(3-2\cdot x)\cdot h-h^{2}+3\cdot x+x^{2}-3\cdot x}{h}$

$f'(x) = \frac{(3-2\cdot x)\cdot h-h^{2}}{h}$

$f'(x) = 3-2\cdot x -h$

If $h \longrightarrow 0$, then:

$f'(x) = 3-2\cdot x$