How do you find f'(x) using the definition of a derivative f(x)=−3x2+6?

The defintion of the derivative isf′(x)=limh→0[f(x+h)−f(x)h]so we have thatf′(x)=limh→0−3(x+h)2+6+3x2−6h⇒f′(x)=limh→0−3h(h+2x)h⇒f′(x)=limh→0−3(h+2x)⇒f′(x)=−6x

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High School
Calculus and Analysis
Calculus 1
Derivatives

Braden Hatfield

Answered question

2022-04-18

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

f (x) = - 3 x^{2} + 6

Answer & Explanation

dizzydevila6cm

Beginner2022-04-19Added 10 answers

The defintion of the derivative is

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

so we have that

f^{'} (x) = lim_{h \to 0} \frac{- 3 {(x + h)}^{2} + 6 + 3 x^{2} - 6}{h} \Rightarrow f^{'} (x) = lim_{h \to 0} \frac{- 3 h (h + 2 x)}{h} \Rightarrow f^{'} (x) = lim_{h \to 0} - 3 (h + 2 x) \Rightarrow f^{'} (x) = - 6 x

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