Question

How do you find the derivation of $f\left( x \right)={{x}^{2}}$ using first principle ?

Answer 1

In the given question you have been asked to find the derivative of a function using first principle. First principle is nothing; it is just finding the first derivative of a function by taking the limit zero. The first derivative of any function can be represented as $\dfrac{dy}{dx}$ or $f'x$ or $y'$. Derivation from first principles is also known as the delta method.

Formula used:
The differentiation of a function using first principle,
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$

Complete step by step answer:
We have the given function,
$f\left( x \right)={{x}^{2}}$
Formula for derivation using first principle:
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$
Replace the value of $x$ to $x+h$, we get
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{{{\left( x+h \right)}^{2}}-{{\left( x \right)}^{2}}}{h}$
Simplifying the above function by using property i.e.${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$, we obtain
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{{{x}^{2}}+2xh+{{h}^{2}}-{{x}^{2}}}{h}$

Combining the like terms, we get
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{2xh+{{h}^{2}}}{h}$
Taking $h$ as a common number, we get
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{h\left( 2x+h \right)}{h}$
Simplifying the above function, we get
$\Rightarrow f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\left( 2x+h \right)$
Putting the value of $h=0$, we get
$\therefore f'x=2x$

Therefore, $f'x=2x$ is the required solution.

Note: When we are asked to find the derivative using first principle, we will have to use the basic formula of differentiation i.e. $f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\dfrac{f\left( x+h \right)-f\left( x \right)}{h}$. First principle method is the most basic method to differentiate any function. The first derivative of any function can be represented as $\dfrac{dy}{dx}$ or $f'x$ or $y'$. In the formula, we have to replace the value of $x$ to $x+h$. Lastly, if we get the value in terms of $h$, then we have to put the value of $h$ equal to zero i.e. the limit value. For differentiating any function using first principle, all we need to know is the basic differentiation formula and after that you have to perform mathematical operations such as addition, subtraction, multiplication and division.