It depends upon your type of Work:
Maths is required if you are working in an applied Science role, i.e. you try to experiments with the known things in Hand i.e. try word embeddings, may be with CNN, and see if the results are good or NOT.
On the other hand , a lot of maths is required if you want to end up as a research scientist, example finding new ways to represent word embedding, or improve the existing word embeddings itself, in case of Text mining.
On the other hand, If you are working as a Software engineer in Machine Learning OR a Machine Learning Engineer, then you just need to train the models using existing knowledge of doing things and Tune it for better performance.
There is a trade off between research and Engineering. More towards Research is More Maths, but More towards Engineering is lesser Maths and More on performance of system in Production.
Another Example to explain would be, for chat Bots.
Research Scientist with Maths background require to write a paper for a new models like how LSTM works and can be Used.
A applied scientiest will try out a business problem like building a chat bot with LSTM first and publish papers how it worked for them in Labs.
A Machine Learning Engineer will replicate the concept which the Applied Scientist had published for their engineering Work (i.e. need to understand maths of paper and replicate it in code, that's it.)
Hope this helps with respect to requirement of knowledge of Maths in Machine Learning
Best Answer
The second reference you give is, in my opinion, still the best book on NN, even though it might be a bit outdated and does not deal with more recent developments like deep architectures. You will get the basics right, and become familiar with all the basic concepts around machine learning.
If you go through the book, you will need linear algebra, multivariate calculus and basic notions of statistics (conditional probabilities, bayes theorem and be familiar with binomial distributions). At some points it deals with calculus of variations. The appendix on calculus of variations should be enough though.