Given the equation \( x^{3}+y^{7}=4 \), find \( \frac{d y}{d x} \) by implicit differentiation.

To find \( \frac{d y}{d x} \) using implicit differentiation, we start with the equation: \[ x^{3} + y^{7} = 4. \] Now, differentiate both sides with respect to \( x \): \[ \frac{d}{dx}(x^{3}) + \frac{d}{dx}(y^{7}) = \frac{d}{dx}(4). \] This gives us: \[ 3x^{2} + 7y^{6} \frac{d y}{d x} = 0. \] Now, isolating \( \frac{d y}{d x} \): \[ 7y^{6} \frac{d y}{d x} = -3x^{2}. \] Dividing both sides by \( 7y^{6} \): \[ \frac{d y}{d x} = -\frac{3x^{2}}{7y^{6}}. \] Thus, the derivative \( \frac{d y}{d x} \) is \[ \frac{d y}{d x} = -\frac{3x^{2}}{7y^{6}}. \]