Firstly the magnitude of rotor current, secondly the flux which interact with the rotor of three phase induction motor and is responsible for producing emf in the rotor part of induction motor, lastly the power factor of rotor of the three phase induction motor.
Combining all these factors together we get the equation of torque as-

Where, T is the torque produced by induction motor,
φ is flux responsible of producing induced emf,
I
2 is rotor current,
cosθ
2 is the power factor of rotor circuit.
The flux φ produced by the stator is proportional to stator emf E
1.
i.e φ ∝ E
1
We know that transformation ratio K is defined as the ratio of secondary voltage (rotor voltage) to that of primary voltage (stator voltage).

Rotor current I
2 is defined as the ratio of rotor induced emf under running condition , sE
2 to total impedance, Z
2 of rotor side,

and total impedance Z
2 on rotor side is given by ,

Putting this value in above equation we get,

We know that power factor is defined as ratio of resistance to that of impedance. The power factor of the rotor circuit is

Putting the value of flux φ, rotor current I
2, power factor cosθ
2 in the equation of torque we get,

Combining similar term we get,

Removing proportionality constant we get,

Where n
s is synchronous speed in r. p. s, n
s = N
s / 60. So, finally the equation of torque becomes,

Derivation of K in torque equation.
In case of three phase induction motor, there occur copper losses in rotor. These rotor copper losses are expressed as
P
c = 3I
22R
2
We know that rotor current,

Substitute this value of I
2 in the equation of rotor copper losses, P
c. So, we get

The ratio of P
2 : P
c : P
m = 1 : s : (1 - s)
Where P
2 is the rotor input,
P
c is the rotor copper losses,
P
m is the mechanical power developed.

Substitute the value of Pc in above equation we get,

On simplifying we get,

The mechanical power developed P
m = Tω,

Substituting the value of P
m

We know that the rotor speed N = N
s(1 - s)
Substituting this value of rotor speed in above equation we get,

N
s is speed in revolution per minute (rpm) and n
s is speed in revolution per sec (rps) and the relation between the two is

Substitute this value of N
s in above equation and simplifying it we get

Comparing both the equations, we get, constant K = 3 / 2πn
s
If we want to find the maximum value of some quantity then we have to differentiate that quantity with respect to some variable parameter and then put it equal to zero. In this case we have to find the condition for maximum torque so we have to differentiate torque with respect to some variable quantity which is slip, s in this case as all other parameters in the equation of torque remains constant.
Now differentiate the above equation by using division rule of differentiation. On differentiating and after putting the terms equal to zero we get,
Substituting the value of this slip in above equation we get the maximum value of torque as,