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f(x) = 83 - x2

Difference Quotient

You are going to plug in f(x) into:

f '(x) lim Δx -> 0 = f(x + Δx) - f(x) / Δx

= 83 - (x + Δx)2 - (83 - x2) / Δx

= 83 - (x2 + 2xΔx + (Δx)2) - 83 + x2 / Δx

= 83 - x2 - 2xΔx - (Δx)2 - 83 + x2 / Δx

The 83 and x2 cancel out leaving:

= - 2xΔx - (Δx)2 / Δx

Through factoring out the Δx:

= Δx (-2x - Δx) / Δx

The Δx cancel out leaving:

= -2x - Δx

The limit of Δx is 0, thus plug in 0 and you get

= -2x - 0

= -2x

Power Rule

The power rule is simply to bring down the exponent and subtract one from the remaining exponent. Changes are done in bold. Reminder: the derivative of a constant is zero.

f'(x) = 0 - (2)x2-1

= -2x

Lvl 15

y = 83 - x^(2)

The n its derivative is

dy/dx = 0 - 2x

dy/dx = -2x

Q: What is the 83 - x2 derivative?

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