How do you find f'(x) using the definition of a derivative for f(x)=6x+2x?

The derivative isf′(x)=limh→0f(x+h)−f(x)h=limh→06(x+h)+2x+h−6x−2xh=limh→06+2⋅x+h−xh=6+2limh→0x+h−xh=6+2limh→0hh(x+h+x)=6+2⋅12x=6+1x

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

Zane Decker

2022-04-15

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

f (x) = 6 x + 2 \sqrt{x}

maggionmoo

Beginner2022-04-16Added 16 answers

The derivative is

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

lim_{h \to 0} 6 + 2 \cdot \frac{\sqrt{x + h} - \sqrt{x}}{h} = 6 + 2 lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} = 6 + 2

lim_{h \to 0} \frac{h}{h (\sqrt{x + h} + \sqrt{x})} = 6 + 2 \cdot \frac{1}{2 \sqrt{x}} = 6 + \frac{1}{\sqrt{x}}

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