Question

Need help with this calc question plz

Answer 1

Given that

`7x^5 - 6xy + 6y^2 = 272`

differentiating both sides with respect to x (using the power, product, and chain rules) yields

`35x^4 - 6x(\mathrm dy)/(\mathrm dx) - 6y + 12y(\mathrm dy)/(\mathrm dx) = 0`

Solving for dy/dx gives

`(\mathrm dy)/(\mathrm dx) = (6y-35x^4)/(12y-6x)`

This gives the slope of the tangent line to the curve at any point (x, y). In particular, the slope of the tangent to (2, 4) is

`(\mathrm dy)/(\mathrm dx)(2,4) = (6\cdot4-35\cdot2^4)/(12\cdot4-6\cdot2) = \boxed{-\frac{134}9}`