Question

How‌ ‌do‌ ‌you‌ ‌find‌ ‌the‌ ‌derivative‌ ‌of‌ ‌${{x}^{{{\log‌ ‌}_{5}}\left(‌ ‌x‌ ‌\right)}}$?‌

Answer 1

To solve the above stated question we need to first change ${{\log }_{5}}$to ${{\log }_{e}}$ which is none other than ln, we will change it to ln as we know the derivation of ln and not of log with any base, to make it easier we will do that. Now to further proceed we will use the chain rule which is basically if the function f(x) is equal to g(h(x)) we will first differentiate g(h(x)) then we will differentiate h(x) and the final result will be the multiplication of both i.e.

& f(x)=g(h(x)) \\\ & \Rightarrow f'(x)=\left[ g'(h(x)) \right]\left[ h'(x) \right] \\\ \end{aligned}$$ This will then result in the required answer. **Complete step by step solution:** In the above question we are going to first change $${{\log }_{5}}$$to $${{\log }_{e}}$$ and we can do this by following the basic principle of logarithm to change the of a logarithm function there are some rules that should hold true are the value of the base should be greater positive and should not be equal to 1 and the logarithmic argument should also be positive so as to change the base of the logarithm. Now to change the base we will follow this procedure: $${{\log }_{a}}b=\dfrac{{{\log }_{x}}b}{{{\log }_{x}}a}$$ Now by applying this in our main function we can get it as: $$f(x)={{x}^{\dfrac{{{\log }_{e}}x}{{{\log }_{e}}5}}}$$ Now we will be applying the chain rule so as to get the differentiated value of the given function in the question. By applying the chain rule we will get the differentiated value as: $$\Rightarrow f'(x)=\left( \dfrac{{{\log }_{e}}x}{{{\log }_{e}}5} \right)\left( {{x}^{\dfrac{{{\log }_{e}}x}{{{\log }_{e}}5}-1}} \right)\left[ \dfrac{1}{x\left( {{\log }_{e}}5 \right)} \right]$$ Now in the above statement we used the differentiated properties that were used were $${{x}^{n}}=n{{x}^{n-1}}$$ and $${{\log }_{e}}x=\dfrac{1}{x}$$ the first property was used first as the function stated was in power of x now by chain rule when we differentiated the power of x the numerator of the function was only the differentiable function as the denominator did not depend upon function x so in this we will use the next derivative property as mentioned above to differentiate the power of x with which we got the final answer. **The derivative of the function given in the question is $$f'(x)=\left( \dfrac{{{\log }_{e}}x}{{{\log }_{e}}5} \right)\left( {{x}^{\dfrac{{{\log }_{e}}x}{{{\log }_{e}}5}-1}} \right)\left[ \dfrac{1}{x\left( {{\log }_{e}}5 \right)} \right]$$** **Note:** In the above question the basic error that is being done are not seeing the base of the logarithmic function, if the base of the logarithmic function is not e then it cannot be differentiated the same way we differentiate logarithmic function with base e, so always make sure while derivation that the base of the logarithmic function is e.