Use logarithmic differentiation to find the derivative of the function f(x) = \frac{ {x}^{ (2)/(5) } \sqrt{ {x}^(2) + 1} }{ \sqrt[5]{ {x}^(4) + 1 } }

Question

Use logarithmic differentiation to find the derivative of the function
`f(x) = \frac{ {x}^{ (2)/(5) } \sqrt{ {x}^(2) + 1} }{ \sqrt[5]{ {x}^(4) + 1 } }`

Answer 1

we are given the following function:

`f(x)=\frac{x^{(2)/(5)}\sqrt[]{x^2+1}}{\sqrt[5]{x^4+1}}`

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}}`

Now we will use the following property of logarithms:

`\ln (a)/(b)=\ln a-\ln b`

applying that property we get:

`\ln f(x)=\ln x^{(2)/(5)}\sqrt[]{x^2+1}-\ln \sqrt[5]{x^4+1}`

Now we will use the following property of logarithms:

`\ln ab=\ln a+\ln b`

applying that property we get:

`\ln f(x)=\ln x^{(2)/(5)}+\ln \sqrt[]{x^2+1}-\ln \sqrt[5]{x^4+1}`

Now we'll make use of the following property of logarithms:

`\ln a^b=b\ln a`

We'll rewrite the roots first as fractional exponents:

`\ln f(x)=\ln x^{(2)/(5)}+\ln (x^2+1)^{(1)/(2)}-\ln (x^4+1)^{(1)/(5)}`

Now we apply the property:

`\ln f(x)=(2)/(5)\ln x+(1)/(2)\ln (x^2+1)-(1)/(5)\ln (x^4+1)`

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

`(d)/(dx)(\ln f(x))=(f^(\prime)(x))/(f(x))`

Applying that to the equation:

`(f^(\prime)(x))/(f(x))=(2)/(5)((1)/(x))+(1)/(2)((2x)/(x^2+1))-(1)/(2)((4x^3)/(x^4+1))`

Simplifying:

`(f^(\prime)(x))/(f(x))=(2)/(5x)+(x)/(x^2+1)-(2x^3)/(x^4+1)`

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

`f^(\prime)(x)=f(x)((2)/(5x)+(x)/(x^2+1)-(2x^3)/(x^4+1))`

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))`

And this is the derivative of the function.