How do you find f'(x) using the limit definition given 4x2−1?

f′(x)=limh→04(x+h)2−1−(4x2−1)hf′(x)=limh→04(x2+2xh+h2)−1−4x2+1hf′(x)=limh→04x2+8xh+4h2−1−4x2+1hf′(x)=limh→08xh+4h2hf'(x)=limh→0h(8x+4h)hf'(x)=8x+4(0)f'(x)=8xThe derivative is therefore f'(x)=8x

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

High School
Calculus and Analysis
Calculus 1
Derivatives

Vanessa Mccarty

Answered question

2022-04-15

How do you find f'(x) using the limit definition given

4 x^{2} - 1

Answer & Explanation

pobijedi6wro

Beginner2022-04-16Added 15 answers

$f^{'} (x) = lim_{h \to 0} \frac{4 {(x + h)}^{2} - 1 - (4 x^{2} - 1)}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{4 (x^{2} + 2 x h + h^{2}) - 1 - 4 x^{2} + 1}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{4 x^{2} + 8 x h + 4 h^{2} - 1 - 4 x^{2} + 1}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{8 x h + 4 h^{2}}{h}$
$f' (x) = \lim_{h \to 0} \frac{h (8 x + 4 h)}{h}$
f'(x)=8x+4(0)
f'(x)=8x
The derivative is therefore f'(x)=8x

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