How do I find the gradient of a function like $f(\vec{v})$ where $$ f(\vec{v}) = \max_{\vec{t}\geq 0} g(\vec{v},\vec{t})$$
For example, I have a function defined as follows:
\begin{align} f(\vec{v}) = \max_{\vec{t} \geq 0} \left( \gamma v_1 \frac{\lambda_A\cdot(1-\exp(-\lambda_At_0))}{\lambda_A + \lambda_B+\lambda_C+\lambda_D} + \gamma v_2 \frac{\lambda_B\cdot(1-\exp(-\lambda_Bt_1))}{\lambda_A + \lambda_B+\lambda_C+\lambda_D} \\ + (c\cdot(1-\exp(-\lambda_Ct_0)) + \gamma v_0) \frac{\lambda_C}{\lambda_A + \lambda_B+\lambda_C+\lambda_D} \\ (c\cdot(1-\exp(-\lambda_Dt_1)) + \gamma v_0) \frac{\lambda_D}{\lambda_A + \lambda_B+\lambda_C+\lambda_D}\right) \end{align} where all $\lambda$ and are positive constants, $c$ is a negative constant, $\gamma \in [0,1]$, and $\exp$ is the exponential function.
I've noticed that this function is linear with respect to $v$, but I'm not entirely sure what to do with that.