Pertanyaan

Jika diberikan fungsi f ( x ) = cos 4 ( x − 1 x + 1 ) maka nilai dari f'(x) = ...

Jika diberikan fungsi $f (x) = cos^{4} (\frac{x + 1}{x - 1})$ maka nilai dari f'(x) = ...

$begin mathsize 12px style 8 over open parentheses x minus 1 close parentheses squared sin cubed open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses sin open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$
$begin mathsize 12px style 8 over open parentheses x minus 1 close parentheses squared cos to the power of 4 open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses sin open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$
$begin mathsize 12px style 8 over open parentheses x minus 1 close parentheses squared cos cubed open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses cos open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$
$begin mathsize 12px style cos cubed open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses sin open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$
$begin mathsize 12px style 8 over open parentheses x minus 1 close parentheses squared cos cubed open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses sin open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$

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Pembahasan

Misalkan:

$begin mathsize 12px style y equals cos to the power of 4 open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$

Misalkan: $begin mathsize 12px style z equals open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses rightwards arrow y equals cos to the power of 4 space z end style$

$begin mathsize 12px style z equals fraction numerator x plus 1 over denominator x minus 1 end fraction table row cell right curly bracket u end cell row cell right curly bracket v end cell end table u equals x plus 1 semicolon space u apostrophe equals 1 v equals x minus 1 semicolon space x apostrophe equals 1 fraction numerator d z over denominator d x end fraction equals fraction numerator u apostrophe v minus u v apostrophe over denominator v squared end fraction equals fraction numerator 1 open parentheses x minus 1 close parentheses minus open parentheses x plus 1 close parentheses 1 over denominator open parentheses x minus 1 close parentheses squared end fraction fraction numerator d z over denominator d x end fraction equals negative 2 over open parentheses x minus 1 close parentheses squared y equals c o s to the power of 4 space z rightwards arrow fraction numerator d y over denominator d z end fraction equals negative 4 space sin space z space cos cubed space z f apostrophe open parentheses x close parentheses equals fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d z end fraction fraction numerator d z over denominator d x end fraction f apostrophe open parentheses x close parentheses equals open parentheses negative 4 space sin space z space cos cubed space z close parentheses open parentheses negative 2 over open parentheses x minus 1 close parentheses squared close parentheses f apostrophe open parentheses x close parentheses equals 8 over open parentheses x minus 1 close parentheses squared sin open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses cos cubed open parentheses fraction numerator x plus 1 over denominator x minus 1 end fraction close parentheses end style$