2019-10-19T09:00:55-04:00
Differentiate from first principle square root of (x+√x).
1
2019-10-21T10:28:34-0400
f ( x ) = x + x f(x)=\sqrt{x+\sqrt{x}} f ( x ) = x + x
f ′ ( x ) = ( x + x ) ′ = lim h → 0 f ( x + h ) − f ( x ) h = f'(x)=\bigg(\sqrt{x+\sqrt{x}}\bigg)'=\lim\limits_{h\rarr0}{f(x+h)-f(x) \over h}= f ′ ( x ) = ( x + x ) ′ = h → 0 lim h f ( x + h ) − f ( x ) =
= lim h → 0 x + h + x + h − x + x h = =\lim\limits_{h\rarr0}{\sqrt{x+h+\sqrt{x+h}}-\sqrt{x+\sqrt{x}} \over h}= = h → 0 lim h x + h + x + h − x + x =
= lim h → 0 x + h + x + h − x + x h ⋅ x + h + x + h + x + x x + h + x + h = =\lim\limits_{h\rarr0}{\sqrt{x+h+\sqrt{x+h}}-\sqrt{x+\sqrt{x}} \over h}\cdot{\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}} \over \sqrt{x+h+\sqrt{x+h}}}= = h → 0 lim h x + h + x + h − x + x ⋅ x + h + x + h x + h + x + h + x + x =
= lim h → 0 x + h + x + h − x − x h ( x + h + x + h + x + x ) = =\lim\limits_{h\rarr0}{ x+h+\sqrt{x+h}-x-\sqrt{x}\over h(\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}})}= = h → 0 lim h ( x + h + x + h + x + x ) x + h + x + h − x − x =
= lim h → 0 h h ( x + h + x + h + x + x ) + =\lim\limits_{h\rarr0}{ h\over h(\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}})}+ = h → 0 lim h ( x + h + x + h + x + x ) h +
+ lim h → 0 x + h − x h ( x + h + x + h + x + x ) = +\lim\limits_{h\rarr0}{ \sqrt{x+h}-\sqrt{x}\over h(\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}})}= + h → 0 lim h ( x + h + x + h + x + x ) x + h − x =
= 1 x + 0 + x + 0 + x + x + ={ 1\over \sqrt{x+0+\sqrt{x+0}}+\sqrt{x+\sqrt{x}}}+ = x + 0 + x + 0 + x + x 1 +
+ lim h → 0 x + h − x h ( x + h + x + h + x + x ) ⋅ x + h + x x + h + x = +\lim\limits_{h\rarr0}{ \sqrt{x+h}-\sqrt{x}\over h(\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}})}\cdot{\sqrt{x+h}+\sqrt{x} \over \sqrt{x+h}+\sqrt{x}}= + h → 0 lim h ( x + h + x + h + x + x ) x + h − x ⋅ x + h + x x + h + x =
= 1 2 x + x + lim h → 0 x + h − x h ( x + h + x + h + x + x ) ( x + h + x ) = ={ 1\over 2\sqrt{x+\sqrt{x}}}+\lim\limits_{h\rarr0}{ x+h-x\over h(\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}})(\sqrt{x+h}+\sqrt{x})}= = 2 x + x 1 + h → 0 lim h ( x + h + x + h + x + x ) ( x + h + x ) x + h − x =
= 1 2 x + x + lim h → 0 1 ( x + h + x + h + x + x ) ( x + h + x ) = ={ 1\over 2\sqrt{x+\sqrt{x}}}+\lim\limits_{h\rarr0}{ 1\over (\sqrt{x+h+\sqrt{x+h}}+\sqrt{x+\sqrt{x}})(\sqrt{x+h}+\sqrt{x})}= = 2 x + x 1 + h → 0 lim ( x + h + x + h + x + x ) ( x + h + x ) 1 =
= 1 2 x + x + 1 ( x + 0 + x + 0 + x + x ) ( x + 0 + x ) = ={ 1\over 2\sqrt{x+\sqrt{x}}}+{ 1\over (\sqrt{x+0+\sqrt{x+0}}+\sqrt{x+\sqrt{x}})(\sqrt{x+0}+\sqrt{x})}= = 2 x + x 1 + ( x + 0 + x + 0 + x + x ) ( x + 0 + x ) 1 =
= 1 2 x + x ( 1 + 1 2 x ) ={ 1\over 2\sqrt{x+\sqrt{x}}}\big(1+{1 \over 2\sqrt{x}}\big) = 2 x + x 1 ( 1 + 2 x 1 )
( x + x ) ′ = 1 2 x + x ( 1 + 1 2 x ) \bigg(\sqrt{x+\sqrt{x}}\bigg)'={ 1\over 2\sqrt{x+\sqrt{x}}}\bigg(1+{1 \over 2\sqrt{x}}\bigg) ( x + x ) ′ = 2 x + x 1 ( 1 + 2 x 1 )
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#340153 on Dec 2023
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