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 \(x\).