DEFINISI FUNGSI TURUNAN

NAMA : NUCKE AURELIA
KELAS : XI MIPA 1

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$\large f(x) = (tan\sqrt[]{x})^{3} \\ f(x+h) = (tan\sqrt[]{x})^{3}\\ \\ f^{'}(x) = \lim_{h -> 0}\ \frac{f(x+h) - f(x)}{h}\\ \lim_{h -> 0}\ \frac{( tan\sqrt[]{x+h} )^{3} - ( tan\sqrt[]{x} )^{3} }{h}\\ \lim_{h -> 0}\ \frac{( tan\sqrt[]{x+h} - tan\sqrt[]{x} )( \ (tan\sqrt[]{x+h})^{2} + tan\sqrt[]{x+h} \ . \ tan\sqrt[]{x} + (tan\sqrt[]{x})^{2} \ ) }{h}\\$ $\large \lim_{h -> 0}\ \frac{ tan ( \ \sqrt[]{x+h} - \sqrt[]{x}\ )( \ 1 + tan\sqrt[]{x+h} \ . \ tan\sqrt[]{x} \ )(tan\sqrt[]{x+h})^{2} \ + tan\sqrt[]{x+h} \ . \ tan\sqrt[]{x} \ + ( \ tan\sqrt[]{x})^{2} \ }{h}\\$ $\large \lim_{h -> 0}\ \frac{ tan(\sqrt[]{x+h} - \sqrt{x} \ . \ \frac{ \sqrt{x+h} + \sqrt{x} }{ \sqrt{x+h} + \sqrt{x} } ) ( 1 + tan\sqrt{x+h} \ . \ tan\sqrt{x})(tan\sqrt{x+h})^{2} + tan\sqrt{x+h} \ . \ tan\sqrt{x} + (tan\sqrt{x})^{2} \ }{h}\\$ $\large \lim_{h -> 0}\ \frac{ tan \frac{ x+h-x }{ \sqrt{x+h} + \sqrt{x} } ( 1 + tan\sqrt{x+h} \ . \ tan\sqrt{x})(tan\sqrt{x+h})^{2} + tan\sqrt{x+h} \ . \ tan\sqrt{x} + (tan\sqrt{x})^{2} \ }{h}\\$ $\large \lim_{h -> 0}\ \frac{ \frac{ \mathbf{tan \ h} }{ \sqrt{x+h} + \sqrt{x} } ( 1 + tan\sqrt{x+h} \ . \ tan\sqrt{x})(tan\sqrt{x+h})^{2} + tan\sqrt{x+h} \ . \ tan\sqrt{x} + (tan\sqrt{x})^{2} \ }{\mathbf{h}}\\$

$= \frac{1}{\sqrt{x+0} + \sqrt{x}}(1+tan\sqrt{x+0} \ . \ tan\sqrt{x})(tan\sqrt{x+0})^{2}+tan\sqrt{x+0} \ . \ tan\sqrt{x}+(tan\sqrt{x})^{2}$
$= \frac{1}{2 \sqrt{x}}( 1+tan^{2}\sqrt{x} )( tan\sqrt{x} )^{2}+( tan\sqrt{x} )^{2}+( tan\sqrt{x} )^{2}$
$= \frac{ sec^{x} }{ 2\sqrt{x} } \ . \ 3 tan^{2}\sqrt{x}$

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DEFINISI FUNGSI TURUNAN