Question

Y is a differentiable function of x. Choose the alternative that is the derivative dy / dx.

$x^3-y^3=1$

a) $x$
b) $3x^2$
c) $\sqrt{3x^2}$
d) $x^2 / y^2$
e) $\frac{3x^2-1}{y^2}$

Answer 1

Differentiating both sides of

$x^3-y^3=1$

with respect to x yields (using the chain rule)

$3x^2 - 3y^2 \dfrac{\mathrm dy}{\mathrm dx} = 0$

Solve for dy/dx :

$3x^2 - 3y^2 \dfrac{\mathrm dy}{\mathrm dx} = 0 \\\\ 3y^2\dfrac{\mathrm dy}{\mathrm dx} = 3x^2 \\\\ \dfrac{\mathrm dy}{\mathrm dx} = \dfrac{3x^2}{3y^2} = \dfrac{x^2}{y^2}$

The answer is then D.