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

$35x^4 - 6x\dfrac{\mathrm dy}{\mathrm dx} - 6y + 12y\dfrac{\mathrm dy}{\mathrm dx} = 0$

Solving for dy/dx gives

$\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{6y-35x^4}{12y-6x}$

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

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