The dot product differentiation rule is $(\vec f(t) \cdot \vec g(t))' = \vec f\ '(t) \cdot \vec g(t) + \vec f(t) \cdot \vec g\ '(t)$, which simplifies to $$(\vec f(t) \cdot \vec f(t))' = 2(\vec f\ '(t) \cdot \vec f(t))$$ when we plug a single vector in for both $\vec f(t)$ and $\vec g(t)$. For example, we find the same answer of $36t^3 + 2t + 4$ whether we plug $\begin{bmatrix}t + 2 \\ 3t^2\end{bmatrix}$ into the LHS or the RHS of the simplified equation. However, when the vectors are functions, I'm having trouble applying this to inner product differentiation.
Let's use the inner product $\int_{-1}^1 f(t)g(t)\ dt$ and the vector $2t^3 + 10$. It seems to me that we should find $(\int_{-1}^1 (2t^3 + 10)(2t^3 + 10)\ dt)' = 2\int_{-1}^1 (6t^2)(2t^3 + 10)\ dt$, by matching the form of the dot product analog. Unfortunately, this LHS must be $0$ (independent of the choice of vector), since the definite integral it contains must return a constant. The RHS is more interesting and works out to $80$, which is more reasonable. Why does this calculation fail, and what is the correct way to migrate the dot product example into an inner product one?