Is d(sinx) and sin(dx) same?

Question

Is d(sinx) and sin(dx) same? If it's not then is there a way to represent sin(dx) or for that matter any function f(x) where x is put to be some differential? Examples are most welcome if needed.

Answer 1

Certainly the way physicists write math you can use both expressions but they are not at all the same. $d\ \sin(x)=\cos(x)\ dx$ is an expression you might use during a $u-$ substitution in an integral. $\sin(dx)$ is the sine of a very small angle. Based on the Taylor series, $\sin(dx)=dx$. Based on your comment, if you have $\sin(x+dx)$ you can expand it as the sine of the sum of two angles, getting $\sin(x)\cos(dx)+\cos(x)\sin(dx)=$(if you only keep first order terms)$\sin(x)+\cos(x)\ dx$. It represents how much $\sin (x)$ changes with $x$ depending on the value of $x$.

Answer 2

$\text{d}(sinx)=sin(x+\text{d}x)-sin(x)=\text{cos}(x)\text{d}x, dx \to 0$.

They aren't same always except when $\text{cos}x=1 \Rightarrow x=2mπ$.