Question

use the definition for the function above to find the derivative.
f(x)=lim_{delta x
ightarrow0}\frac{7-(x + delta x)^2-(\text{ }-x^2)}{delta x}
=lim_{delta x
ightarrow0}\frac{7 - x^2-(\text{ })xdelta x-(delta x)^2-(\text{ })+x^2}{delta x}
=lim_{delta x
ightarrow0}\frac{(\text{ })xdelta x-(delta x)^2}{delta x}
=lim_{delta x
ightarrow0}((\text{ })x-delta x)
f(x)=\text{ }

Explanation:

Step1: Identify f(x)

f(x) = 7 - x², so first box is 7.

Step2: Expand (x+Δx)²

(x+Δx)² = x² + 2xΔx + (Δx)², so second boxes are 2,7.

Step3: Simplify numerator

Cancel terms: -2xΔx - (Δx)², third box is -2.

Step4: Factor and divide by Δx

Δx(-2x - Δx)/Δx = -2x - Δx, fourth box is -2.

Step5: Evaluate limit

lim(Δx→0)(-2x - Δx) = -2x.

Answer:

-2x