## Summary

The limit of a function exists and is equal to \(L\) if the values of \(f(x)\) get closer to \(L\) from both sides as \(x\) gets closer to \(a\).

\[\lim_{x\to a} f(x) = L\]

**Average gradient or average rate of change:**\[\text{Average gradient } = \cfrac{f(x+h)-f(x)}{h}\]

**Gradient at a point or instantaneous rate of change:**\[f'(x) = \lim_{h\to 0}\cfrac{f(x+h)-f(x)}{h}\]

**Notation**\[{f}'(x)={y}’=\cfrac{dy}{dx}=\cfrac{df}{dx}=\cfrac{d}{dx}[f(x)]=Df(x)={D}_{x}y\]

**Differentiating from first principles:**\[f'(x) = \lim_{h\to 0}\cfrac{f(x+h)-f(x)}{h}\]

**Rules for differentiation:**General rule for differentiation:

\[\cfrac{d}{dx}[{x}^{n}]=n{x}^{n-1}, \text{ where } n \in \mathbb{R} \text{ and } n \ne 0.\]

The derivative of a constant is equal to zero.

\[\cfrac{d}{dx}[k]= 0\]

The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

\[\cfrac{d}{dx}[k \cdot f(x) ]=k \cfrac{d}{dx}[ f(x) ]\]

The derivative of a sum is equal to the sum of the derivatives.

\[\cfrac{d}{dx}[f(x)+g(x)]=\cfrac{d}{dx}[f(x) ] + \cfrac{d}{dx}[g(x)]\]

The derivative of a difference is equal to the difference of the derivatives.

\[\cfrac{d}{dx}[f(x) – g(x)]=\cfrac{d}{dx}[f(x) ] – \cfrac{d}{dx}[g(x)]\]

**Second derivative:**\[f”(x) = \cfrac{d}{dx}[f'(x)]\]

**Sketching graphs:**The gradient of the curve and the tangent to the curve at stationary points is zero.

Finding the stationary points: let \(f'(x) = 0\) and solve for \(x\).

A stationary point can either be a local maximum, a local minimum or a point of inflection.

**Optimisation problems:**Use the given information to formulate an expression that contains only one variable.

Differentiate the expression, let the derivative equal zero and solve the equation.