Differentiation is used to find the gradients of curves at specific points on the curve.

The differentiated form of a function is known as its first derivative. Differentiating again gives the second derivative.

If the original function is $\bm{y}$, then the first derivative is $\bm{\dfrac{dy}{dx}}$ and the second derivative is $\bm{\dfrac{d^2y}{dx^2}}$.

If the original function is $\bm{f(x)}$, then the first derivative is $\bm{f'(x)}$ and the second derivative is $\bm{f''(x)}$.

The first derivative $f'(x)$ is the gradient of the tangent to the graph of $y=f(x)$ at a general point $(x,y)$. The first derivative is also known as the gradient function.

The first derivative $\Big(\dfrac{dy}{dx}\Big)$ is the rate of change of $y$ with respect to $x$.

The second derivative $\Big(\dfrac{d^2y}{dx^2}\Big)$ is the rate of change of the first derivative $\Big(\dfrac{dy}{dx}\Big)$ with respect to $x$.