You don't need to derive backpropagation by hand, but you do need to know why it works.
There's a recurring debate in AI education about how much math you actually need to be productive. The pragmatist camp says: none, use the libraries. The purist camp says: you need to understand everything from first principles. Both positions miss something. You don't need to derive the chain rule on a whiteboard to train a neural network with PyTorch — but you absolutely need to understand what a gradient is, why vanishing gradients are a problem, and what learning rate schedules are doing, or you will be unable to diagnose why your model is failing to converge. The practical answer is: understand the concepts, not the derivations.
The place where calculus knowledge pays the most immediate dividends in applied AI is not training — it's debugging. When a loss function plateaus, when gradients explode, when a model oscillates instead of converging, you need enough calculus intuition to form a hypothesis about what's happening and where to look. Engineers who lack that foundation treat optimizer hyperparameters as magic knobs to turn, which is an expensive and slow way to work. Anecdotally, the engineers who progress fastest in ML roles are those who can read a loss curve and form a mechanistic hypothesis, not just those who know the most APIs.
The most efficient entry point to calculus-for-AI is 3Blue1Brown's Essence of Calculus series, which builds the right intuitions geometrically rather than algebraically. Pair it with the specific backpropagation lecture from Andrej Karpathy's micrograd series, and you'll have everything you need to reason about gradient-based learning without grinding through a calculus textbook.