Question 1154661
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

given: {{{y *sin(16x) = x *cos( 2y)}}}, ({{{pi/2}}},{{{ pi/4}}})


{{{y *sin (16x) = x* cos (2y)}}}

{{{16y *cos(16x) }}}+ {{{y}}}'{{{sin(16x) }}}={{{ cos(2y) }}}- {{{2xy}}}'{{{sin(2y)}}}


To find the slope {{{m}}}, replace {{{y}}}' with {{{m}}}, {{{x}}} with {{{pi/2}}}, and {{{y }}}with {{{pi/4}}}.

{{{16*(pi/4)* cos(16(pi/2)) + m*sin(16(pi/2) )= cos(2(pi/4)) - 2(pi/2)*m*sin(2(pi/4))}}}

{{{4pi* cos(8pi) + m*sin(8pi) = cos(pi/2) - pi*m*sin(pi/2)}}}

{{{4pi *1 + m*0 = 0 - pi*m*1}}}

{{{4pi = -pi*m}}}

{{{m = -4}}}

Equation of the tangent line:

{{{y - pi/4 = -4(x - pi/2)}}}

{{{y = -4x + 2pi + pi/4}}}

{{{y = -4x + 9pi/4}}}