Finding the tenth derivative of $f(x) = e^x\sin x$ at $x=0$ I came across this Question where I have to find $$f^{(10)}$$ for the following function at $x = 0$
$$f(x) = e^x\sin x$$
I tried differentiating a few times to get a pattern but didn’t get one, can someone provide the solution.

One trick here is to use $e^{ix}=\cos x+i\sin x$ and define $g(x)=e^x\cos x$ then $f(x)$ and $g(x)$ are both real functions. 
Let $h(x)=g(x)+if(x)=e^{(1+i)x}$ then the tenth derivative of $h(x)$ is $(1+i)^{10}h(x)$ and the tenth derivative of $f(x)$ is the imaginary part of this.
Because you only want the value at $x=0$ you can evaluate there, with $h(0)=g(0)+if(0)$
