What is the gradient of a non-linear SVM with respect to the input?

Question

The objective function of an SVM is the following:

$$J(\mathbf{w}, b)=C \sum_{i=1}^{m} \max \left(0,1-y^{(i)}\left(\mathbf{w}^{t} \cdot \mathbf{x}^{(i)}+b\right)\right)+\frac{1}{2} \mathbf{w}^{t} \cdot \mathbf{w}$$ where

• $\mathbf{w}$ is the model's feature weights and $b$ is its bias parameter
• $\mathbf{x}^{(i)}$ is the $i^\text{th}$ training instance's feature vector
• $y^{(i)}$ is the target class ($-1$ or $1$) for the $i^\text{th}$ instance
• $m$ is the number of training instances
• $C$ is the regularisation hyper-parameter

And if I was to use a kernel, this would become:

$$J(\mathbf{w}, b)=C \sum_{i=1}^{m} \max \left(0,1-y^{(i)}\left(\mathbf{u}^{t} \cdot \mathbf{K}^{(i)}+b\right)\right)+\frac{1}{2} \mathbf{u}^{t} \cdot \mathbf{K} \cdot \mathbf{u}$$

where the kernel can be the Gaussian kernel:

$$K(\mathbf{u}, \mathbf{v})=e^{-\gamma\|\mathbf{u}-\mathbf{v}\|^{2}}$$

How would I go about finding its gradient with respect to the input?

I need to know this as to then apply this to a larger problem of a CNN with its last layer being this SVM, so I can then find the gradient of this output wrt the input of the CNN.