Rules for Calculating Derivatives and Differentials for Functions of a Single Variable
Constant Rule: The Derivative and Differential of any Constant Value is always 0 (since it does not change).
Differential Rule: The Differential of Any Variable is denoted by Preceeding the Name of the Variable with \(d\).
For example, the Differential of Variable \(x\) is denoted by \(dx\). Similarly, the Differential of Variable \(t\) is denoted by \(dt\).
Identity Rule: The Derivative of Any Variable with respect to itself is always \(1\).
Function Rule: Given any Function \(F\) of a Single Variable \(x\) as follows
\(F=f(x)\)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) (denoted by \(F'\)) is given as follows
where \(dF\) is the Differential of the Function \(F\). This Differential is calculated as the Product of the Derivative of the Function \(F\) and the Differential of the Variable \(x\) as follows
\(dF= F'dx=f'(x)dx={\Large \frac{df(x)}{dx}}dx\)
Absolute Value Rule: Given any Function \(F\) which is Absolute Value of a Function of a Single Variable \(x\) as follows
\(F=|f(x)|\)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) is given as the Absolute Value of the Derivative of the Function as
\(F'={\Large \frac{dF}{dx}}=|f'(x)|\)
and the Differential of the Function \(F\) is calculated as
\(dF= F'dx=|f'(x)|dx\)
Sum Rule: Given any Function \(F\) which is a Sum of 2 or More Functions of a Single Variable \(x\) as follows
\(F=f_1(x) + f_2(x) + \cdots + f_n(x)\)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) is given as the Sum of Derivatives of those Functions as
Please note that if All the Functions \(f_1(x), f_2(x), ..., f_n(x)\) are Same then the Product Rule is same as the Constant Power Rule
(i.e. the Constant Power Rule can be derived from the Product Rule).
Quotient Rule: Given any Function \(F\) which is a Quotient of 2 Functions of a Single Variable \(x\) as follows
\(F={\Large \frac{f_n(x)}{f_d(x)}}=f_n(x){f_d(x)}^{-1}\) (where \(f_n\) is the Numerator Function and \(f_n\) is the Denominator Function)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) is given as
and the Differential of the Function \(F\) is calculated as
\(dF=F'dx={\Large \frac{f'(x)}{f(x)}}dx\)
Logarithm Rule for Any Constant Base: Given any Function \(F\) which is a Log of a Function of a Single Variable \(x\) with Any Constant as Base as follows
Please note that Logarithm Rule for Base \(e\) is a Special Case for this Rule when \(a=e\).
Logarithm Rule for Any Function Base: Given any Function \(F\) which is a Log of a Function of a Single Variable \(x\) with Any Other Function of Single Variable \(x\) as Base as follows
Please note that Logarithm Rule for Any Constant Base is a Special Case for this Rule when \(f_b(x)\) is a Constant.
Rule for Function Powers of \(e\): Given any Function \(F\) which is \(e\) Raised to the Power of a Function of a Single Variable \(x\) as follows
\(F=e^{f(x)}\)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) is given as
\(F'={\Large \frac{dF}{dx}}=e^{f(x)}f'(x)\)
and the Differential of the Function \(F\) is calculated as
\(dF=F'dx=e^{f(x)}f'(x)dx\)
Rule for Function Powers of Any Constant: Given any Function \(F\) which is Any Constant Raised to the Power of a Function of a Single Variable \(x\) as follows
\(F=a^{f(x)}\)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) is given as
\(F'={\Large \frac{dF}{dx}}=a^{f(x)}f'(x)\ln(a)\)
and the Differential of the Function \(F\) is calculated as
\(dF=F'dx=a^{f(x)}f'(x)\ln(a)dx\)
Please note that Rule for Function Powers of \(e\) is a Special Case for this Rule when \(a=e\).
Rule for Function Powers of Any Function: Given any Function \(F\) which is a Function of a Single Variable \(x\) Raised to the Power of Any Other Function of Single Variable \(x\) as follows
\(F=f_b(x)^{f_e(x)}\)
then the Derivative of the Function \(F\) with respect to the Variable \(x\) is given as
Please note that Rule for Function Powers of Any Constant is a Special Case for this Rule when \(f_b(x)\) is a Constant.
Derivative of a Function with respect to another Function: Given 2 Functions \(F_1\) and \(F_2\) of a Single Variable \(x\) as follows
\(F_1=f_1(x)\), \(F_2=f_2(x)\)
then the Derivative of one Function with respect to the other is calculated as follows
Derivative of \(F_1\) with respect to \(x\) = \(F_1'={\Large \frac{dF_1}{dx}}=f_1'(x)\)
Derivative of \(F_2\) with respect to \(x\) = \(F_2'={\Large \frac{dF_2}{dx}}=f_2'(x)\)
\(\therefore\) Derivative of \(F_1\) with respect to \(F_2\) = \({\Large \frac{F_1'}{F_2'}}={\Large \frac{f_1'(x)}{f_2'(x)}}\)
\(\therefore\) Derivative of \(F_2\) with respect to \(F_1\) = \({\Large \frac{F_2'}{F_1'}}={\Large \frac{f_2'(x)}{f_1'(x)}}\)
The Number of Times the Derivative is Calculated for a Function is called the Order of the Derivative of the Function.
The Derivative of Function \(F\) of a Single Variable \(x\) called its First Derivative (or First Order Derivative)
and is denoted by \({\Large \frac{dF}{dx}}\) or \(F'\) or \(F^1\). The Derivative of Derivative of Function \(F\) is also called its Second Derivative (or Second Order Derivative)
and denoted by \({\Large \frac{d^2F}{dx^2}}\) or \(F''\) or \(F^2\). The Derivative of Derivative of Derivative of Function \(F\) is also called its Third Derivative (or Third Order Derivative)
and denoted by \({\Large \frac{d^3F}{dx^3}}\) or \(F'''\) or \(F^3\) and so on. Similarly, \(n^{th}\) Derivative of Function \(F\) (or \(n^{th}\) Order Derivative) is denoted by \({\Large \frac{d^nF}{dx^n}}\) or \(F^n\).
