A curve has the equation 6x^(3/2) + 5y^2 = 2 (a) By differentiating implicitly, find dy/dx in terms of x and y. (b) Hence, find the gradient of the curve at the point (4, 3).

(a) To differentiate implicitly, differentiate x’s as normal and differentiate y’s with respect to y before multiplying by dy/dx. Therefore the differentiating the curve gives
9x^(1/2) + 10y*(dy/dx) = 0
which can be rearranged to give dy/dx = -9x^(1/2) / 10y
(b) at (4, 3) dy/dx = -94^(1/2) / 103
m^(1/2) is equivalent to √m so
dy/dx = -92 / 103 = -3 / 5