we are given the following function:
![f(x)=\frac{x^{(2)/(5)}\sqrt[]{x^2+1}}{\sqrt[5]{x^4+1}}](https://img.qammunity.org/2023/formulas/mathematics/college/ryfegcyfto5if5frz7hbsgs9tmps1vgqm0.png)
we are asked to find the derivative of this function. To do that, we will take logarithms on both sides of the equation, like this:
![\ln f(x)=\ln \frac{x^{(2)/(5)}\sqrt[]{x^2+1}}{\sqrt[5]{x^4+1}}](https://img.qammunity.org/2023/formulas/mathematics/college/51suly89zpx9czwqgp5vqdzxzna3gwda6t.png)
Now we will use the following property of logarithms:

applying that property we get:
![\ln f(x)=\ln x^{(2)/(5)}\sqrt[]{x^2+1}-\ln \sqrt[5]{x^4+1}](https://img.qammunity.org/2023/formulas/mathematics/college/vuluviyl802b1i7xaubtu1j6d22vmys9vv.png)
Now we will use the following property of logarithms:

applying that property we get:
![\ln f(x)=\ln x^{(2)/(5)}+\ln \sqrt[]{x^2+1}-\ln \sqrt[5]{x^4+1}](https://img.qammunity.org/2023/formulas/mathematics/college/rbgn81r40860oakzkat9s9ryv3jvwmyewa.png)
Now we'll make use of the following property of logarithms:

We'll rewrite the roots first as fractional exponents:

Now we apply the property:

Now we differentiate on both sides of the equation, following the rules of logarithmic differentiation, that is:

Applying that to the equation:

Simplifying:

Now we solve for f'(x), like this:

Now we substitute the value of f(x)
![f^(\prime)(x)=\frac{x^{(2)/(5)}\sqrt[]{x^2+1}}{\sqrt[5]{x^4+1}}((2)/(5x)+(x)/(x^2+1)-(2x^3)/(x^4+1))](https://img.qammunity.org/2023/formulas/mathematics/college/juoc3g2xfyckzlydfhf9h36zibtiazgwa8.png)
And this is the derivative of the function.