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 \(\begin{pmatrix}ax^n\end{pmatrix}' = n\times ax^{n1}\). 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 \(f(x) = \frac{x^23}{3sin(x)}\).

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

