Question

Use implicit differentiation to find the slope of the tangent line to the curve y/(x + 4 y)= x^6 − 9
at the point ( 1, [(−8)/33] ).
m = ?

Answer 1

after applying differentiation, work on dy/dx, substitute the given point, the result dy/dx is the slope m of a tangent line to the given point...

Answer 2

the equation on left-hand can be solved using...
\[d \left( \frac{ u }{ v } \right)=\frac{ v du - u dv }{ v^{2} }\]

Answer 3

u = y and du = dy
v = x+4y and dv = dx + 4 dy
v^2 = (x+4y)^2 = x^2+8xy+14y^2
therefore...
\[d \left( \frac{ y }{ x+4y } \right)=\frac{ (x+4y)dy-y(dx+4dy) }{ x^{2}+8xy+16y^{2} }\]
\[=\frac{ xdy+4ydy-ydx-4ydy }{ x^2+8xy+16y^2 }\]
\[=\frac{ xdy-ydx }{ x^2+8xy+16y^2 }\]

Answer 4

for the right-hand side...
\[d(x^6-9)=6x^5dx-0=6x^5dx\]

Answer 5

equating both sides...
\[\frac{ xdy-ydx }{ x^2+8xy+16y^2 }=6x^5dx\]
\[xdy-ydx=(x^2+8xy+16y^2)6x^5dx\]
\[xdy-ydx=6x^7dx+48x^6ydx+96x^5y^2dx\]
\[xdy=(6x^7+48x^6y+96x^5y^2+y)dx\]
\[\frac{ dy }{ dx }=\frac{ 6x^7+48x^6y+96x^5y^2+y }{ x }\]

Answer 6

substitute now the coordinates of the given point... the result is the slope :-)