Let $y=f(x)$ be a real function. If $f(x)$ is differentiable at $x_0$, then the the expression $dy=f'(x_0)dx$ is called the differential of $f$ at $x_0$. Or using the Leibniz's notation for the derivatives
$dy=\left. \frac{df}{dx}\right\vert _{x_{0}}\; dx$
For a generic $x$, we thus have
$dy=f'(x)dx=\frac{df}{dx} dx$
In the picture below this equation in $dx,dy$ represents the tangent line to the graph of $f(x)$ at $x_0$ in the translated coordinates system $dx,dy$. Both $dy$ and $dx$ are interpreted as infinitesimals. The differential $dy$ is approximately the change of $y$ when $x$ changes by an arbitrary small quantity $dx$.

In the present case we have the function $t=f(x)=\sin x$, whose derivative is $t^{\prime }=f^{\prime }(x)=\frac{df}{dx}=\cos x.$ So, the diferential $dt$ is
$dt=f^{\prime }(x)dx=\frac{df}{dx}dx=\cos x\;dx.$
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