Question

Differentiate the function.

13. \(y = {\log _8}\left( {{x^2} + 3x} \right)\)

Short answer

The derivative of the function is\(\frac{{2x + 3}}{{({x^2} + 3x)\log 8}}\).

Step-by-step solution

Use the derivative of logarithmic function

Rule 2: The derivative of \(\ln x\) is,

\(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)

Evaluating the derivative of given function

\(\begin{aligned}{c}y &= {\log _8}({x^2} + 3x)\\= \frac{{\log ({x^2} + 3x)}}{{\log 8}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{{\log }_b}a= \frac{{\log a}}{{\log b}}} \right)\end{aligned}\)

Differentiating y with respect to x,

\(\begin{aligned}{c}y &= \frac{1}{{\log 8}}\log ({x^2} + 3x)\\\frac{{dy}}{{dx}}&= \frac{1}{{\log 8}}\frac{{d\log ({x^2} + 3x)}}{{dx}}\\\frac{{dy}}{{dx}} &= \frac{1}{{\log 8}}\frac{1}{{({x^2} + 3x)}}\frac{{d\left( {{x^2} + 3x} \right)}}{{dx}}\\\frac{{dy}}{{dx}} &= \frac{1}{{\log 8}}\frac{1}{{({x^2} + 3x)}}\left( {2x + 3} \right)\\\frac{{dy}}{{dx}} &= \frac{{2x + 3}}{{\left( {{x^2} + 3x} \right)\log 8}}\end{aligned}\)

\(\)Thus, the value of the derivative is \(\frac{{2x + 3}}{{\left( {{x^2} + 3x} \right)\log 8}}\) .