81
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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
-1/x2
-4/x2
The antiderivative of x2 + x is 1/3x3 + 1/2x2 + C.
0
All it means to take the second derivative is to take the derivative of a function twice. For example, say you start with the function y=x2+2x The first derivative would be 2x+2 But when you take the derivative the first derivative you get the second derivative which would be 2