# Use logarithmic differentiation to find the derivative of the function. y = sqrt((x - 4)/(x^8 + 2)).

## Question:

Use logarithmic differentiation to find the derivative of the function.

{eq}y = \sqrt{\frac{x - 4}{x^8 + 2}} {/eq}

## Logarithmic Differentiation

It is a method to differentiate the function by employing the logarithm derivative of a function. In this method, we take the log both sides and then differentiate.

{eq}\Rightarrow \ y=\sqrt{\frac{(x-4)}{(x^{8}+2)}}\\ \text{taking log both sides}\\ \Rightarrow \ \ln(y)=\frac{1}{2}(\ln(x-4)-\ln(x^{8}+2))\\ \text{diiferentiate with respect to x}\\ \Rightarrow \ \frac{1}{y}y'=\frac{1}{2}(\frac{1}{(x-4)}-\frac{8x^{7}}{(x^{8}+2)})\\ \Rightarrow \ y'=y\frac{1}{2}(\frac{1}{(x-4)}-\frac{8x^{7}}{(x^{8}+2)})\\ \Rightarrow \ y'=\frac{1}{2}\sqrt{\frac{(x-4)}{(x^{8}+2)}}(\frac{1}{(x-4)}-\frac{8x^{7}}{(x^{8}+2)})\\ {/eq}