Newton invented calculus so that he could do physics.
Not quite. Newton is given the credit for inventing calculus in the same way that Einstein is given the credit for inventing relativistic thought, even though he piggy backed on Lorentz (not downplaying the leap in logic that Einstein took and the work he did--just trying to show that he stood on the shoulders of those who came before him). Much as calculus can be defined as a description of how things change, the calculus itself is something that changed over time.
Going all the way back to the ancient Greek mathematicians, we have Eudoxus and Archimedes using the method of exhaustion to approximate limits and integrals, with the latter using the method to calculate the area of a circle. But if we want to talk about Newton and what he is given credit for, there are a few heavy hitters we need to talk about first.
Galileo (and his contemporaries) performed experiments and thought experiments in which they discovered that the acceleration of a body due to gravity is invariant. This is where the second law of motion came from, which was already well known to Newton--it was a given used by many. His leap was in applying it to inverse square motion, though even then it's likely that he got the inverse square law from Hooke and took off from that perch to apply it to the cosmos.
As far as what we think of when we think of calculus, the credit again does not really go to Newton. If anything, the credit should go to his advisor, Isaac Barrow, who provided the first proof for the Fundamental Theorem of Calculus. He too built on those who came before him, though. What Newton really did was see how derivatives and integrals can be used under one big umbrella.
So, Newton didn't really invent calculus. He gobbled together a lot of things that other people did, then applied them with a slightly different bent. His major contributions were in the inverse square motion (from which he explained tidal friction, precession, how gravity decreases with altitude, etc.), the third law of motion (which implies conservation of momentum and angular momentum), and his ingenious techniques for solving really difficult problems.
But we give Newton credit for inventing calculus and the laws of motion because he is an extremely important figure in physics, and it is easier to just give him credit for things he described while describing the stuff he actually did invent, even if most of them were well known at the time and not novel.
And by the way, the notation we use in calculus today came from Liebniz.
