Question

How do you find the derivative using limits of $f\left( x \right) = 3x + 2$?

Answer 1

A number that a function reaches as the independent variable of the function reaches a given value is known as a limit. In simple words, a limit is defined as a value that a function reaches the output for the given set of input values. For a real function $f\left( x \right)$ and real number ‘$k$’, the limit is represented by: $\mathop {\lim }\limits_{x \to k} f\left( x \right) = z$. In order to solve the given question, we should recall that the derivative of any function $f\left( x \right)$can be given by the formula: $f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}} \right)$.

Complete step by step solution:
Given is$f\left( x \right) = 3x + 2$. We have to find the derivative using the limits. We know that the derivative of any function $f\left( x \right)$can be given by $f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}} \right)$.
Let $f\left( x \right) = 3x + 2$.
So, $f\left( {x + h} \right) = 3\left( {x + h} \right) + 2$-----(1)
Taking out the derivative of$f\left( x \right) = 3x + 2$.
Using (1) in the derivative formula we get,
$f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{3\left( {x + h} \right) + 2 - \left( {3x + 2} \right)}}{h}} \right)$
Now, using the distributive law of multiplication, which says that when a factor is multiplied by the sum of two terms, then it is essential to multiply each of the two terms by the factor and then perform the addition operation.
$\mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{3x + 3h + 2 - 3x - 2}}{h}} \right) \\\ \Rightarrow\mathop {\lim }\limits_{h \to 0} \left( {\dfrac{{3h}}{h}} \right) \\\ \Rightarrow\mathop {\lim }\limits_{h \to 0} \left( 3 \right) \\\\\ \therefore 3 \\\$
Hence, the derivative of $f\left( x \right) = 3x + 2$ using limits is $3$.

Note: The first derivative of a function is a function which gives instantaneous rate of change of desired function at any point. It can also be the rate of change of distance with respect to the time. The formula of derivative can be used to determine an expression which describes the gradient of the graph. Derivatives are an important tool of calculus.