Differentiation vs. Derivative

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Differentiation is the operation that takes a function and produces its rate-of-change function, often written as $D f$ or $\frac{d}{dx}f(x)$.
The derivative is the result of that operation: the new function $f’(x)$ or, at a specific point, the number $f’(a)$.
As a function, the derivative assigns each $x$ the limit $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ when it exists.
At a point, the derivative $f’(a)$ is the instantaneous rate of change and the slope of the tangent line at $x=a$.
Example: for $f(x)=x^2$, differentiation yields $f’(x)=2x$, and the derivative at $x=3$ is $f’(3)=6$.