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Zorluk: OrtaDifferentiation from First Principles

By evaluating the limit of the difference quotient lim⁡h→0f(x+h)−f(x)h\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, determine the value of the derivative of the function f(x)=2x2−4x+5f(x) = 2x^2 - 4x + 5 at the point where x=3x = 3.

Cevap: 8

Cevap

The value of the derivative of f(x)=2x2−4x+5f(x) = 2x^2 - 4x + 5 at x=3x = 3 is 8.
Applying first principles, f(x+h)−f(x)=4xh+2h2−4hf(x+h) - f(x) = 4xh + 2h^2 - 4h. Dividing by hh gives 4x+2h−44x + 2h - 4. Taking the limit as h→0h \to 0 produces f′(x)=4x−4f'(x) = 4x - 4. Substituting x=3x = 3 yields 4(3)−4=84(3) - 4 = 8.

Adım Adım Çözüm

1
Substitute (x+h)(x+h) into the function f(x)=2x2−4x+5f(x) = 2x^2 - 4x + 5 and expand
f(x+h)=2(x2+2xh+h2)−4x−4h+5=2x2+4xh+2h2−4x−4h+5f(x+h) = 2(x^2 + 2xh + h^2) - 4x - 4h + 5 = 2x^2 + 4xh + 2h^2 - 4x - 4h + 5
Apply algebraic expansion to determine the value of the function at x+hx+h.
2
Form the difference f(x+h)−f(x)f(x+h) - f(x)
f(x+h)−f(x)=4xh+2h2−4hf(x+h) - f(x) = 4xh + 2h^2 - 4h
Subtract the original function terms to leave only terms containing hh.
3
Divide the difference by hh
\frac{f(x+h) - f(x)}{h} = 4x + 2h - 4
Simplify the difference quotient prior to taking the limit.
4
Compute the limit as h→0h \to 0
f′(x)=4x−4f'(x) = 4x - 4
Taking the limit yields the general derivative function f′(x)f'(x).
5
Evaluate f′(x)f'(x) at x=3x = 3
f′(3)=4(3)−4=8f'(3) = 4(3) - 4 = 8
Substitute x=3x = 3 to find the numerical rate of change at the given point.

Anahtar Kavram

Differentiation from First Principles
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Differentiation from First Principles

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