# Summary and Main Ideas

## 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.