Question

The money to be spent for the we fare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) received from the sale of $x$ units of a product is given by $R(x) = 3{x^2} + 36x + 5$ , find the marginal revenue, when $x = 5$, and write value does the question indicate.

Answer 1

Here, we are going to take derivative with respect to x., Also using formula ${\frac{{d({x^n})}}{{dx}} = n{x^{n - 1}}}$

Complete step by step solution:
Given that: The total revenue (in rupees) received from the sale of units of a product is given by $R(x) = 3{x^2} + 36x + 5$
Here, Marginal Revenue = $\frac{{dR}}{{dx}}$
$\begin{array}{l} R(x) = 3{x^2} + 36x + 5\\\ \frac{{dR(x)}}{{dx}} = \frac{{d(3{x^2} + 36x + 5)}}{{dx}} \end{array}$
$\begin{array}{l} \frac{{dR(x)}}{{dx}} = 6x + 36\\\ \\\ \frac{{dR(x)}}{{dx}} = 6(5) + 36\\\ \frac{{dR(x)}}{{dx}} = 30 + 36 \end{array}$ [ when $x = 5$, Marginal Revenue will be]
Marginal Revenue $= 66$

Hence, the required marginal revenue is Rupees 66.

Additional Information: Always remember the five basic differentiation rule.
The constant rule
For Example: Derivative of any constant number is zero.
$\begin{array}{l} f(x) = c\\\ f'(x) = 0 \end{array}$
The power rule
For Example: power is multiplied in this form.

$\begin{array}{l} f(x) = {x^n}\\\ f'(x) = n.{x^{n - 1}} \end{array}$
3.) The constant multiple rule
For Example: Always take constant aside before applying derivative to variables.
$\begin{array}{l} f(x) = 5{x^{^3}}\\\ f'(x) = 5.f'({x^3})\\\ f'(x) = 5 \times 3 \times {x^2}\\\ f'(x) = 15{x^2} \end{array}$
4) The sum rule
For Example: derivative is applied in each term separately.
$\begin{array}{l} f(x) = {x^4} + {x^3} + 2x\\\ f'(x) = f'({x^4}) + f'({x^3}) + 2f'(x) \end{array}$
5) The difference rule
For Example: Derivative is applied in each term separately.
$\begin{array}{l} f(x) = 2{x^4} - 5{x^2} - 8x\\\ f'(x) = 2f'({x^4}) - 5f'({x^2}) - 8f(x) \end{array}$

Note: Always solve by taking consideration of the respective derivatives with the given substitution and simplified form would be the answer.