How do you use the definition of a derivative to find the derivative of #f(x) = 4 + 9x - x^2#?

Question

Ezekiel Alexander · Accepted Answer

#f'(x)=-2x+9#

The limit definition of a derivative states that

#f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h#

Substituting #f(x)=4+9x-x^2# into #f'(x)#,

#f'(x)=lim_(hrarr0)((4+9(x+h)-(x+h)^2)-(4+9x-x^2))/h#

From this point on, you want to expand and simplify.

#f'(x)=lim_(hrarr0)((4+9x+9h-(x^2+2xh+h^2))-(4+9x-x^2))/h#

#f'(x)=lim_(hrarr0)(4+9x+9h-x^2-2xh-h^2-4-9x+x^2)/h#

#f'(x)=lim_(hrarr0)(color(red)cancelcolor(black)4color(blue)cancelcolor(black)(+9x)+9hcolor(teal)cancelcolor(black)(-x^2)-2xh-h^2color(red)cancelcolor(black)(-4)color(blue)cancelcolor(black)(-9x)color(teal)cancelcolor(black)(+x^2))/h#

#f'(x)=lim_(hrarr0)(9h-2xh-h^2)/h#

#f'(x)=lim_(hrarr0)(color(red)cancelcolor(black)h(9-2x-h))/color(red)cancelcolor(black)h#

#f'(x)=lim_(hrarr0)(9-2x-h)#

Plugging in #h=0#,

#f'(x)=(9-2x-0)#

#f'(x)=9-2x#

#color(green)(|bar(ul(color(white)(a/a)color(black)(f'(x)=-2x+9)color(white)(a/a)|)))#

Isabella Ackerman · Answer

undefined To use the definition of a derivative to find the derivative of the function $ f(x) = 4 + 9x - x^2 $:

1. Recall the definition of a derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the given function into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(4 + 9(x + h) - (x + h)^2) - (4 + 9x - x^2)}{h} $$

3. Expand and simplify the expression:
$$ f'(x) = \lim_{h \to 0} \frac{(4 + 9x + 9h - (x^2 + 2xh + h^2)) - (4 + 9x - x^2)}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{4 + 9x + 9h - x^2 - 2xh - h^2 - 4 - 9x + x^2}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{9h - 2xh - h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(9 - 2x - h)}{h} $$

5. Cancel out $ h $ in the numerator and denominator:
$$ f'(x) = \lim_{h \to 0} (9 - 2x - h) $$

6. Evaluate the limit as $ h $ approaches 0:
$$ f'(x) = 9 - 2x $$

Therefore, the derivative of the function $ f(x) = 4 + 9x - x^2 $ is $ f'(x) = 9 - 2x $.