Final answer:
To find dy/dx, differentiate the given equation with respect to x. To find dy/dt, differentiate the equation with respect to t.
Step-by-step explanation:
To find dy/dx, we need to differentiate the given equation y=(t/x⁴)+(x/t) with respect to x. Using the power rule of differentiation, we get:
dy/dx = d/dx [(t/x⁴) + (x/t)]
= ((d/dx)(t/x⁴)) + ((d/dx)(x/t))
= (1/x⁴)(d/dx)(t) + (1/t)(d/dx)(x)
= -4t/x⁵ + 1/t
Therefore, dy/dx = -4t/x⁵ + 1/t.
To find dy/dt, we differentiate the equation with respect to t:
dy/dt = d/dt [(t/x⁴) + (x/t)]
= ((d/dt)(t/x⁴)) + ((d/dt)(x/t))
= (1/x⁴)(d/dt)(t) + (1/t)(d/dt)(x)
= 1/x⁴ - x/t²
Therefore, dy/dt = 1/x⁴ - x/t².