Summary
Previous
Applications of differential calculus

Next
End of chapter exercises

6.8 Summary (EMCHM)

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:
\[\text{Average gradient } = \frac{f\left(x+h\right)f\left(x\right)}{h}\] 
Gradient at a point or instantaneous rate of change:
\[f'(x) = \lim_{h\to 0}\frac{f\left(x+h\right)f\left(x\right)}{h}\] 
Notation
\[{f}'\left(x\right)={y}'=\frac{dy}{dx}=\frac{df}{dx}=\frac{d}{dx}[f\left(x\right)]=Df\left(x\right)={D}_{x}y\] 
Differentiating from first principles:
\[f'(x) = \lim_{h\to 0}\frac{f\left(x+h\right)f\left(x\right)}{h}\] 
Rules for differentiation:

General rule for differentiation:
\[\frac{d}{dx}\left[{x}^{n}\right]=n{x}^{n1}, \text{ where } n \in \mathbb{R} \text{ and } n \ne 0.\] 
The derivative of a constant is equal to zero.
\[\frac{d}{dx}\left[k\right]= 0\] 
The derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.
\[\frac{d}{dx}\left[k \cdot f\left(x\right) \right]=k \frac{d}{dx}\left[ f\left(x\right) \right]\] 
The derivative of a sum is equal to the sum of the derivatives.
\[\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right) \right] + \frac{d}{dx}\left[g\left(x\right)\right]\] 
The derivative of a difference is equal to the difference of the derivatives.
\[\frac{d}{dx}\left[f\left(x\right)  g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right) \right]  \frac{d}{dx}\left[g\left(x\right)\right]\]


Second derivative:
\[f''(x) = \frac{d}{dx}[f'(x)]\] 
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.
Previous
Applications of differential calculus

Table of Contents 
Next
End of chapter exercises
