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Find the derivative of y = 2x2 − x + 3 from first principles.

Mathematics

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Answer

4x - 1

Graph of y = 2x² − x + 3

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y-intercept03
Minimum0.252.88

Step-by-step solution

5 steps

To find the derivative of y=2x2−x+3y = 2x^2 - x + 3 from first principles, we use the definition:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
  1. Identify f(x)f(x) and f(x+h)f(x+h).

    Given f(x)=2x2−x+3f(x) = 2x^2 - x + 3.
    Substitute (x+h)(x+h) into f(x)f(x) to find f(x+h)f(x+h):

    f(x+h)=2(x+h)2−(x+h)+3f(x+h) = 2(x+h)^2 - (x+h) + 3 f(x+h)=2(x2+2xh+h2)−x−h+3f(x+h) = 2(x^2 + 2xh + h^2) - x - h + 3 f(x+h)=2x2+4xh+2h2−x−h+3f(x+h) = 2x^2 + 4xh + 2h^2 - x - h + 3
  2. Calculate f(x+h)−f(x)f(x+h) - f(x).

    f(x+h)−f(x)=(2x2+4xh+2h2−x−h+3)−(2x2−x+3)f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - x - h + 3) - (2x^2 - x + 3) f(x+h)−f(x)=2x2+4xh+2h2−x−h+3−2x2+x−3f(x+h) - f(x) = 2x^2 + 4xh + 2h^2 - x - h + 3 - 2x^2 + x - 3 f(x+h)−f(x)=4xh+2h2−hf(x+h) - f(x) = 4xh + 2h^2 - h
  3. Substitute into the first principles formula.

    f′(x)=lim⁡h→04xh+2h2−hhf'(x) = \lim_{h \to 0} \frac{4xh + 2h^2 - h}{h}
  4. Factor out hh from the numerator and simplify.

    f′(x)=lim⁡h→0h(4x+2h−1)hf'(x) = \lim_{h \to 0} \frac{h(4x + 2h - 1)}{h}

    Since h→0h \to 0 but h≠0h \neq 0, we can cancel hh:

    f′(x)=lim⁡h→0(4x+2h−1)f'(x) = \lim_{h \to 0} (4x + 2h - 1)
  5. Evaluate the limit by substituting h=0h=0.

    f′(x)=4x+2(0)−1f'(x) = 4x + 2(0) - 1 f′(x)=4x−1f'(x) = 4x - 1

    The derivative of y=2x2−x+3y = 2x^2 - x + 3 is 4x−1\boxed{4x - 1}.

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