Differentiate the following function with respect of `x : x^n\log_a x`

Explanation

Let `u=x^{n} ; v=log _{a} x=frac{log x}{log a}`


Then,`u^{prime}=n x^{n-1} ; v^{prime}=frac{1}{x log a}`
Using the product rule :



`frac{d}{d x}(u v)=u v^{prime}+v u^{prime} `


`frac{d}{d x}(x^{n} log _{a} x)=x^{n} cdot frac{1}{x log a}+log _{a} x(n x^{n-1}) `


`=x^{n-1} frac{1}{log a}+log _{a} x(n x^{n-1}) `


`=x^{n-1}(frac{1}{log a}+n log _{a} x)`