How do you find the derivative of&nbsp;f(x)=2x2+x-1&nbsp;using the limit process?

Based on the limit definition of derivative:dfdx=limh→0f(x+h)−f(x)hdfdx=limh→0(2(x+h)2+x+h−1)−(2x2+x−1)hdfdx=limh→02x2+4xh+2h2+x+h−1−2x2−x+1hdfdx=limh→04xh+2h2+hhdfdx=limh→02h+4x+1dfdx=4x+1

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High School
Calculus and Analysis
Calculus 1
Derivatives

Brooks Barker

Answered question

2022-04-13

How do you find the derivative of $f (x) = 2 x^{2} + x - 1$ using the limit process?

Answer & Explanation

Zaria Lamb

Beginner2022-04-14Added 8 answers

Based on the limit definition of derivative:

\frac{d f}{d x} = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

\frac{d f}{d x} = lim_{h \to 0} \frac{(2 {(x + h)}^{2} + x + h - 1) - (2 x^{2} + x - 1)}{h}

\frac{d f}{d x} = lim_{h \to 0} \frac{2 x^{2} + 4 x h + 2 h^{2} + x + h - 1 - 2 x^{2} - x + 1}{h}

\frac{d f}{d x} = lim_{h \to 0} \frac{4 x h + 2 h^{2} + h}{h}

\frac{d f}{d x} = lim_{h \to 0} 2 h + 4 x + 1

\frac{d f}{d x} = 4 x + 1

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