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Zorluk: KolayDifferentiation from First Principles

Using differentiation from first principles, what is the derivative of the function f(x)=4−x2f(x) = 4 - x^2 with respect to xx?

  1. −2x-2xCevap
  2. B
    2x2x
  3. C
    −2x−1-2x - 1
  4. D
    4−2x4 - 2x

Cevap

−2x-2x
Differentiating from first principles involves finding the limit of f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h} as h→0h \to 0. For f(x)=4−x2f(x) = 4 - x^2, expanding f(x+h)f(x+h) gives 4−x2−2xh−h24 - x^2 - 2xh - h^2. Subtracting f(x)f(x) yields −2xh−h2-2xh - h^2, and dividing by hh gives −2x−h-2x - h. Taking the limit as h→0h \to 0 leaves −2x-2x.

Adım Adım Çözüm

1
Express f(x+h)f(x+h) for f(x)=4−x2f(x) = 4 - x^2
f(x+h)=4−(x+h)2=4−(x2+2xh+h2)=4−x2−2xh−h2f(x+h) = 4 - (x+h)^2 = 4 - (x^2 + 2xh + h^2) = 4 - x^2 - 2xh - h^2
Substitute x+hx+h into the original function definition.
2
Set up the difference f(x+h)−f(x)f(x+h) - f(x)
f(x+h)−f(x)=(4−x2−2xh−h2)−(4−x2)=−2xh−h2f(x+h) - f(x) = (4 - x^2 - 2xh - h^2) - (4 - x^2) = -2xh - h^2
Subtract f(x)f(x) to find the net change in yy.
3
Divide the difference by hh to form the difference quotient
\frac{f(x+h) - f(x)}{h} = \frac{-2xh - h^2}{h} = -2x - h
Divide each term in the numerator by hh.
4
Take the limit as h→0h \to 0
f′(x)=lim⁡h→0(−2x−h)=−2xf'(x) = \lim_{h \to 0} (-2x - h) = -2x
Evaluate the derivative by letting hh approach zero.

Anahtar Kavram

Differentiation from First Principles
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Konular

Differentiation from First Principles

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