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joja [24]
2 years ago
10

Derivative using direct definition of derivative 2-x/2+x

Mathematics
1 answer:
Daniel [21]2 years ago
3 0

Let <em>f(x)</em> = (2 - <em>x</em>)/(2 + <em>x</em>). By definition of the derivative,

\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h

\displaystyle f'(x)=\lim_{h\to0}\frac{\frac{2-(x+h)}{2+(x+h)}-\frac{2-x}{2+x}}h

\displaystyle f'(x)=\lim_{h\to0}\frac{\frac{(2-x-h)(2+x)-(2-x)(2+x+h)}{(2+x+h)(2+x)}}h

\displaystyle f'(x)=\lim_{h\to0}\frac{-4h}{h(2+x+h)(2+x)}

\displaystyle f'(x)=-4\lim_{h\to0}\frac1{(2+x+h)(2+x)}=\boxed{-\frac4{(2+x)^2}}

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