Given: #xln(y) + 6y = 2x^3#
Differentiate each term with respect to x:
#(d(xln(y)))/dx + (d(6y))/dx = (d(2x^3))/dx#
Use the product rule, #(d(uv))/dx = (du)/dxv + u(dv)/dx#, on the first term, where #u = x# and and #v = ln(y)#:
#(d(xln(y)))/dx = (d(x))/dxln(y) + x(d(ln(y)))/dx#
On the second term, within the product rule, we must use the chain rule:
#(d(xln(y)))/dx = (d(x))/dxln(y) + x(d(ln(y)))/(dy)dy/dx#
#(d(xln(y)))/dx = ln(y) + x/ydy/dx#
Substitute this in place of the first term:
#ln(y) + x/ydy/dx + (d(6y))/dx = (d(2x^3))/dx#
For the next term, we use the chain rule:
#ln(y) + x/ydy/dx + (d(6y))/dy dy/dx = (d(2x^3))/dx#
#ln(y) + x/ydy/dx + 6dy/dx = (d(2x^3))/dx#
For the last term, we use the power rule #(d(x^n))/dx = nx^(n-1)#:
#ln(y) + x/ydy/dx + 6dy/dx = 6x^2#
Subtract #ln(y)# from both sides:
#x/ydy/dx + 6dy/dx = 6x^2- ln(y)#
Factor out #dy/dx#:
#(x/y + 6)dy/dx = 6x^2- ln(y)#
Divide by the leading coefficient:
#dy/dx = (6x^2- ln(y))/(x/y + 6)#
Multiply by #y/y#:
#dy/dx = (6yx^2- yln(y))/(x + 6y)#
Because the point #(1,1)# is not on the curve, I do not recommend that you evaluate the derivative at this point.
