Question

Differentiate. y = log₅(2x³-1)

Answer 1

(1/(ln 5))((6x^2)/(2x^3-1))

Answer 2

Change the base to ln

remmber
$log_a(b)= \frac{log_c(b)}{log_c(a)}$
ln is just base e

so
$log_5(2x^3-1)= \frac{log_e(2x^3-1)}{log_e(5)} = \frac{ln(2x^3-1))}{ln(5)}$

remember that tthe deritivive of $\frac{f(x)}{g(x)}$ is $\frac{f'(x)g(x)-g'(x)f(x)}{g(x)^2}$
so

let's take the deritivies seperately first
deritivive of ln(2x^3-1) is $\frac{6x^2}{2x^3-1}$
deritivive of ln(5) is 0


$\frac{ln(2x^3-1))}{ln(5)} = \frac{( \frac{6x^2}{2x^3-1})(ln(5))-(0)(ln(2x^3-1))}{(ln(5))^2}$=$\frac{( \frac{6x^2}{2x^3-1})(ln(5))}{(ln(5))^2}$=$\frac{ \frac{6x^2}{2x^3-1}}{ln(5)}= \frac{6x^2}{(2x^3-1)(ln(5))}$