How do you find f'(x) using the definition of a derivative for f(x)=x2−1?

Definition of a derivative:lim△x→0f(x+△x)−f(x)△xFilling in your function:lim△x→0(x+△x)2−1−x2+1△x=lim△x→0x2+2x△x+△x2−x2△x=lim△x→02x△x+△x2△x=lim△x→02x+△x=2x

The definition of a derivative is:f′(x)=limh→0f(x+h)−f(x)hHere, f(x)=x2−1f′(x)=limh→0(x+h)2−1−x2+1hf′(x)=limh→0x2+2xh+h2−x2hf′(x)=limh→02xh+h2hf′(x)=limh→02x+h=2x+0=2xf'(x)=2x for f(x)=x2−1

Definition of a derivative:lim△x→0f(x+△x)−f(x)△xFilling in your function:lim△x→0(x+△x)2−1−x2+1△x=lim△x→0x2+2x△x+△x2−x2△x=lim△x→02x△x+△x2△x=lim△x→02x+△x=2x

The definition of a derivative is:f′(x)=limh→0f(x+h)−f(x)hHere, f(x)=x2−1f′(x)=limh→0(x+h)2−1−x2+1hf′(x)=limh→0x2+2xh+h2−x2hf′(x)=limh→02xh+h2hf′(x)=limh→02x+h=2x+0=2xf'(x)=2x for f(x)=x2−1

"How do you find f'(x) using the definition..." was answered by our community

Quinn Dean

Answered question

2022-04-16

How do you find f'(x) using the definition of a derivative for

f (x) = x^{2} - 1

Answer & Explanation

xxkrnjangxxed9q

Beginner2022-04-17Added 14 answers

Definition of a derivative:

lim_{△ x \to 0} \frac{f (x + △ x) - f (x)}{△ x}

Filling in your function:

lim_{△ x \to 0} \frac{{(x + △ x)}^{2} - 1 - x^{2} + 1}{△ x}

= lim_{△ x \to 0} \frac{x^{2} + 2 x △ x + △ x^{2} - x^{2}}{△ x}

= lim_{△ x \to 0} \frac{2 x △ x + △ x^{2}}{△ x}

= lim_{△ x \to 0} 2 x + △ x = 2 x

ze2m1ingkdvu

Beginner2022-04-18Added 16 answers

The definition of a derivative is:

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

Here,

f (x) = x^{2} - 1

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

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

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

f^{'} (x) = lim_{h \to 0} 2 x + h = 2 x + 0 = 2 x

f'(x)=2x for

f (x) = x^{2} - 1

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