# Find the derivative. f(t)=\frac{t^3+t}{\ln\Big(\frac{t^2}{t}\Big)}

## Question:

Find the derivative.

{eq}f(t)=\frac{t^3+t}{\ln\Big(\frac{t^2}{t}\Big)} {/eq}

## Quotient Rule of Differentiation:

The derivative of the ratio between two functions {eq}f(t) {/eq} and {eq}g(t) {/eq}

can be calculated by applying the Quotient Rule of Differentiation

{eq}\displaystyle D\left[ f/g \right] = \frac{f'g -fg' }{ g^2 }. {/eq}

## Answer and Explanation:

The derivative of the function

{eq}\displaystyle f(t)=\frac{t^3+t}{\ln\Big(\frac{t^2}{t}\Big)} =\frac{t^3+t}{\ln(t)} {/eq}

is found by applying the quotient rule of differentiation

{eq}\displaystyle D\left[ g/h \right] = \frac{g'h -gh' }{ h^2 }. {/eq}

As a result, we have that

{eq}\displaystyle f'(t)=\frac{\frac{d}{dt}\left(t^3+t\right)\ln \left(t\right)-\frac{d}{dt}\left(\ln \left(t\right)\right)\left(t^3+t\right)}{\left(\ln \left(t\right)\right)^2} \\ \displaystyle = \frac{\left(3t^2+1\right)\ln \left(t\right)-\frac{1}{t}\left(t^3+t\right)}{\ln ^2\left(t\right)} \\ \displaystyle = \frac{3t^2\ln \left(t\right)+\ln \left(t\right)-t^2-1}{\ln ^2\left(t\right)}. {/eq}

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