Question

Differentiate from first principles for the function $3{\text{x}}$

Answer 1

The first principle is nothing but it is the first derivative of the function. We have the first derivative of function with the limit of zero. We can use a formula for finding the difference from first principles.

Formula used
The formula for differentiate from first principle is,
${\text{f(x)}} = \mathop {\lim }\limits_{{\text{h}} \to 0} \dfrac{{{\text{f(x + h)}} - {\text{f(x)}}}}{{\text{h}}}$
Where,
${\text{x}}$ be the function given in the question

Complete step-by-step answer:
The data given in the question is,
The function given in the question is,
${\text{f(x)}} = 3{\text{x}}$
To find differentiate from first principle of the above given function by using the formula,
Substitute the function in the formula we get,
$\Rightarrow \mathop {\lim }\limits_{{\text{h}} \to 0} \dfrac{{{\text{3(x + h)}} - 3{\text{x}}}}{{\text{h}}}$
By solving the above we get,
$\Rightarrow \mathop {\lim }\limits_{{\text{h}} \to 0} \dfrac{{{\text{3x + 3h}} - 3{\text{x}}}}{{\text{h}}}$
By simplifying the above we get,
$\Rightarrow \mathop {\lim }\limits_{{\text{h}} \to 0} \dfrac{{{\text{3h}}}}{{\text{h}}}$
By cancelling the above we get,
$\Rightarrow \mathop {\lim }\limits_{{\text{h}} \to 0} 3$
At last we get,
$\Rightarrow 3$
The first derivative of the above given function is $3$
$\therefore$ The required first derivative principle of the above given function is $3$.
Hence, the difference from the first principle of the given function is $3$.

Additional information: At last if the value of the function has ${\text{h}}$ then we have to substitute the limit value to that. The derivation of any constant will be equal to zero otherwise we can say it as the derivative of any whole number is equal to zero.

Note: The first derivative can be represented as $\dfrac{{dy}}{{dx}}$ or ${{\text{y}}^{'}}$ . In the formula, in place of ${\text{h}}$ no need to substitute anything. We need to substitute ${\text{x + h}}$ in place of ${\text{x}}$ . Differentiation from first principles is also known as delta method.