Rules for Calculating Derivatives and Differentials for Functions of a Single Variable
Constant Rule: As per Constant Rule, Derivative and Differential of any Constant Value is always 0 (since it does not change).
Identity Rule: As per Identity Rule, the Derivative of any Variable \(x\) with respect to itself is always \(1\). The Differential of Variable \(x\) is denoted by \(dx\).
Function Rule: As per Function Rule , if any Function \(F\) of Variable \(x\) given as follows
\(F=f(x)\)
then the Derivative of the Function \(F\) with Respect to the Variable \(x\) (denoted by \(F'\) or \(f'(x)\)) is given as follows
\(F'=\frac{dF}{dx}=f'(x)= \frac{d(f(x))}{dx}\)
where \(dF\) or \(d(f(x))\) 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
\(dF= d(f(x))= F'dx=f'(x)dx\)
Power Rule: The Power Rule is used to calculate the Derivatives and Differentials of a Function that is given as a Real Number Powers of a Variable.
For example, if a Function \(F\) is given as Real Number Power of Variable \(x\) as follows
\(F=x^n\) (where \(n\) is any Real Number)
then as per Power Rule the Derivative of the Function \(F\) with Respect to the Variable \(x\) is given as
\(F'=\frac{dF}{dx}=n*x^{n-1}\)
Hence, as per Function Rule , the Differential of the Function \(F\) (denoted by \(dF\)) is calculated as
\(dF= F'dx = n*x^{n-1}*dx \)
Sum Rule: The Sum Rule is used to calculate the Derivatives and Differentials of a Function that is given as a Sum of 2 or More Functions of a Variable.
For example, if a Function \(F\) is given as Sum of Functions of Variable \(x\) as follows
\(F=f_1(x) + f_2(x) + \cdots + f_n(x)\)
then as per Sum Rule the Derivative of the Function \(F\) with Respect to the Variable \(x\) is given as
So, as per Sum Rule, if a Function is given as Sum of 2 or More Functions of a Variable, then its Derivative/Differential with respect to that Variable is given as Sum of Derivative/Differential of those Functions.
Product Rule: The Product Rule is used to calculate the Derivatives and Differentials of a Function that is given as a Product of 2 or More Functions of a Variable.
For example, if a Function \(F\) is given as Product of Functions of Variable \(x\) as follows
\(F=f_1(x) * f_2(x) * \cdots * f_n(x)\)
then as per Product Rule the Derivative of the Function \(F\) with Respect to the Variable \(x\) is given as
Quotient Rule: The Quotient Rule is used to calculate the Derivatives and Differentials of a Function that is given as a Quotient of 2 Functions of a Variable.
For example, if a Function \(F\) is given as Quotient of Functions of Variable \(x\) as follows
\(F=\frac{f_1(x)}{f_2(x)}\)
then as per Quotient Rule the Derivative of the Function \(F\) with Respect to the Variable \(x\) is given as
Chain Rule: The Chain Rule is used to calculate the Derivatives and Differentials of a Function that is given as Nested Functions of a Variable.
For example, if a Function \(F\) is given as follows
\(F=f_n(f_{n-1}(f_{n-2}(...(f_2(f_1(x)))))\)
then as per Chain Rule the Derivative of the Function \(F\) with Respect to the Variable \(x\) is given as
\(F'=\frac{dF}{dx}=\)Derivative of \(f_n\) WRT \(f_{n-1}\) * Derivative of \(f_{n-1}\) WRT \(f_{n-2}\) * \(\cdots\) * Derivative of \(f_2\) WRT \(f_1\) * Derivative of \(f_1\) WRT \(x\)
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 it is possible to calculate the Derivative of one Function with respect to the other as follows
Derivative of \(F_1\) with respect to \(x\) = \(F_1'=\frac{dF_1}{dx}=\frac{d(f_1(x))}{dx}\)
Derivative of \(F_2\) with respect to \(x\) = \(F_2'=\frac{dF_2}{dx}=\frac{d(f_2(x))}{dx}\)
\(\therefore\) Derivative of \(F_1\) with respect to \(F_2\) = \(\frac{F_1'}{F_2'}=\frac{\frac{dF_1}{dx}}{\frac{dF_2}{dx}}=\frac{\frac{d(f_1(x))}{dx}}{\frac{d(f_2(x))}{dx}}\)
\(\therefore\) Derivative of \(F_2\) with respect to \(F_1\) = \(\frac{F_2'}{F_1'}=\frac{\frac{dF_2}{dx}}{\frac{dF_1}{dx}}=\frac{\frac{d(f_2(x))}{dx}}{\frac{d(f_1(x))}{dx}}\)
