# Calculus

## Differentiation

### Differentiation Techniques

Topics Description
What's a derivative? Differentiation from First Principles
Power Rule for Differentiation We learn how to differentiate any function that can be expressed as a power of $$x$$. The power rule for differentiation is presented as $$\begin{pmatrix}ax^n\end{pmatrix}' = n\times ax^{n-1}$$. This is seen for the three cases:
• $$n$$ a positive integer
• $$n$$ a fraction
• $$n$$ a negative integer
Operations with Derivatives
Product Rule for Differentiation We learn how to differentiate functions that can be expressed as the product of two functions. The formula, a tutorial, as well as worked examples and exercises will help in learning the technique. By the end of this section we'll know, and be comfortable using, the formula $$\begin{pmatrix}u(x).v(x)\end{pmatrix}' = u'(x).v(x)+u(x).v'(x)$$. For example, by the end of this section we'll know how to differentiate $$f(x) = 3x^2.sin(x)\}). Quotient Rule for Differentiation We learn how to differentiate functions that can be expressed as the quotient of two functions, that's one function being divided by another. The formula, a tutorial, as well as worked examples and exercises will help in learning the technique. By the end of this section we'll know, and be comfortable using, the formula \(\begin{pmatrix}\frac{u(x)}{v(x)}\end{pmatrix}' = \frac{u'(x).v(x) - u(x).v'(x)}{\begin{bmatrix}v(x)\end{bmatrix}^2}$$. For example, by the end of this section we'll know how to differentiate $$f(x) = \frac{x^2-3}{3sin(x)}$$.
Chain Rule for Differentiation We learn how to differentiate functions that are composite functions, those are functions embedded inside other functions. For example, by the end of this section we'll know how to differentiate $$f(x)=3.sin\begin{pmatrix}x^2+1\end{pmatrix}$$.
Implicit Differentiation

### Analyzing Curves with Differentiation

Topics Description
Tangents and Normals to Curves We learn a two step-by-step methods to learn how to find the equation of a tangent and how to find the equation of a normal to a curve at a point$$P$$ along its length.
Stationary Points - Finding Maximum, Minimum & Horizontal Points of Inflexion We learn how to find the coordinates of stationary points along a curve's length by solving $$\frac{dy}{dx} = 0$$. The technique is clearly explained with a step-by-step method as well as tutorials, worked examples and exercises with answer keys.