Answer:
[tex]8x - h[/tex]
Step-by-step explanation:
We are solving for the difference quotient, or the input to the limit in the definition of the derivative, for the function:
[tex]f(x) = -4x^2 + 3x - 5[/tex]
We are given the difference quotient value to be:
[tex]\dfrac{f(x+h) - f(x)}{h}[/tex]
Plugging [tex](x + h)[/tex] into [tex]f(x)[/tex] gives us:
[tex]f(x + h) = -4(x+h)^2+3(x+h)-5[/tex]
We can expand and combine like terms to get:
[tex]-4(x^2+2xh+h^2)+3x+3h-5[/tex]
[tex]=-4x^2 - 8xh - 4h^2 + 3x + 3h - 5[/tex]
[tex]=-4x^2 - ((8-3)h + 3)x - h^2 + 3h - 5[/tex]
[tex]=-4x^2 - (5h + 3)x - h^2 + 3h - 5[/tex]
Now, we can replace [tex]f(x+h)[/tex] in the difference quotient with this expression:
[tex]\dfrac{f(x+h) - f(x)}{h} = \dfrac{\left[\frac{}{}-\!\!4x^2 - (5h + 3)x - h^2 + 3h - 5\frac{}{}\right] - \left[\frac{}{}-4x^2 + 3x - 5\frac{}{}\right]}{h}[/tex]
Simplifying and combining like terms, we get:
[tex]\dfrac{-4x^2 - (5h + 3)x - h^2 + 3h - 5 + 4x^2 - 3x + 5}{h}[/tex]
[tex]=\dfrac{(-4 + 4)x^2 - (5h + 3h + 3 - 3)x - h^2 + (-5 + 5)}{h}[/tex]
[tex]=\dfrac{8hx - h^2}{h}[/tex]
[tex]= \dfrac{h(8x - h)}{h}[/tex]
[tex]\boxed{= 8x - h}[/tex]
