`y = log_5sqrt(x^2-1)` Find the derivative of the function

`y=log_5 sqrt(x^2-1)`

Before taking the derivative of the function, express the radical in exponent form.

`y = log_5 (x^2-1)^(1/2)`

Then, apply the logarithm rule ` log_b (a^m) = m* log_b(a)` .

`y = 1/2log_5 (x^2-1)`

From here, proceed to take the derivative of the function. Take note that the derivative formula of logarithm is `d/dx [log_b (u)] = 1/(ln(b) * u) * (du)/dx` .

Applying this formula, the derivative of the function will be:

`(dy)/dx = d/dx[1/2log_5 (x^2-1)]`

`(dy)/dx = 1/2 d/dx[log_5 (x^2-1)]`

`(dy)/dx = 1/2 *1/(ln(5)*(x^2-1)) * d/dx(x^2-1)`

`(dy)/dx = 1/2 *1/(ln(5)*(x^2-1)) *2x`

`(dy)/dx = (2x)/(2(x^2-1)ln(5))`

`(dy)/dx = x/((x^2-1)ln(5))`

Therefore, the derivative of the function is `(dy)/dx = x/((x^2-1)ln(5))` .