What is the derivative of this?

I have a function of the following form:

$J = \|W^TW-I\|_F^2$

Where, $W$ is a matrix and $F$ is the Frobenius Norm.

How can I find the derivative of $\frac{\partial J}{\partial W}$ ?

You can use the product rule for matrix differentiation, $$ \frac{\partial\mathrm{trace}(f(X)G(X))}{\partial X} = \left. \frac{\partial\mathrm{trace}(f(X)G(Y))}{\partial X}+\frac{\partial\mathrm{trace}(f(Y)G(X))}{\partial X}\right|_{Y=X} $$ along with the simpler identity $$ \frac{\partial\mathrm{trace}(AX)}{\partial X}=A^\top. $$

For your problem this would go as follows. First write the problem in this form: \begin{eqnarray*} J &=& \|W^\top W-I\|_F^2 \\ &=& \mathrm{trace}((W^\top W-I)^\top(W^\top W-I))\\ &=& \mathrm{trace}(W^\top WW^\top W - 2W^\top W + I) \end{eqnarray*}

The derivative can now be computed much as for a scalar function. First we split it up into two parts \begin{eqnarray*} J'(W) &=& \frac{\partial}{\partial W}\mathrm{trace}(W^\top WW^\top W - 2W^\top W + I)\\ &=& \frac{\partial}{\partial W}\mathrm{trace}(W^\top WW^\top W) - 2\frac{\partial}{\partial W}\mathrm{trace}(W^\top W)\\ &=& J_1'(W) -2 J_2'(W) \end{eqnarray*} Starting with the simplest term we have \begin{eqnarray*} J_2'(W) &=& \frac{\partial}{\partial W}\mathrm{trace}(W^\top W)\\ &=& \left. \frac{\partial}{\partial W}\mathrm{trace}(W^\top Y + Y^\top W)\right|_{Y=W}\\ &=& \left.Y + (Y^\top)^\top\right|_{Y=W}\\ &=& 2W \end{eqnarray*} And the quartic term: \begin{eqnarray*} J_1'(W) &=& \frac{\partial}{\partial W}\mathrm{trace}(W^\top WW^\top W)\\ &=& \left. \frac{\partial}{\partial W}\mathrm{trace}(W^\top W Y^\top Y + Y^\top Y W^\top W)\right|_{Y=W}\\ &=& 2\left. \frac{\partial}{\partial W}\mathrm{trace}(W^\top W Y^\top Y)\right|_{Y=W}\\ &=& 2\left. \frac{\partial}{\partial W}\mathrm{trace}(W^\top Y Y^\top Y + Y^\top W Y^\top Y )\right|_{Y=W}\\ &=& \left. 2 Y Y^\top Y + 2(Y^\top Y Y^\top ) ^\top\right|_{Y=W}\\ &=& 4 W W^\top W \end{eqnarray*} So the derivative of $J$ should be $$ J'(W) = 4 W W^\top W - 4 W = 4 W(W^\top W-I). $$