How do you find f'(x) using the definition of a derivative #f(x) = -3x^2 + 6#?

The defintion of the derivative is

so we have that

#f'(x)=lim_(h->0)[-3(x+h)^2+6+3x^2-6]/h=> f'(x)=lim_(h->0)[-3h(h+2x)]/h=> f'(x)=lim_(h->0)-3(h+2x)=> f'(x)=-6x#

To find ( f'(x) ) using the definition of a derivative for the function ( f(x) = -3x^2 + 6 ), you use the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute the given function ( f(x) = -3x^2 + 6 ) into the formula:

[ f'(x) = \lim_{h \to 0} \frac{-3(x + h)^2 + 6 - (-3x^2 + 6)}{h} ]

Expand and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{-3(x^2 + 2hx + h^2) + 6 + 3x^2 - 6}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-3x^2 - 6hx - 3h^2 + 6 + 3x^2 - 6}{h} ] [ f'(x) = \lim_{h \to 0} \frac{-6hx - 3h^2}{h} ] [ f'(x) = \lim_{h \to 0} -6x - 3h ] [ f'(x) = -6x ]

So, the derivative of the function ( f(x) = -3x^2 + 6 ) is ( f'(x) = -6x ).