The value of $ \frac{d}{dx}[{{x}^{n}}\,{{\log }_{a}}\,x{{e}^{x}}] $ is
- $ {{e}^{x}}\,{{\log }_{a}}\,x+\frac{{{x}^{n-1}}}{{{\log }_{e}}\,a} $
- $ {{e}^{x}}{{x}^{n-1}}\,\left\{ x{{\log }_{a}}\,x+\frac{1}{{{\log }_{e}}\,a}+n\,{{\log }_{a}}x \right\} $
- $ n{{x}^{n-1}}\,{{\log }_{a}}\,x{{e}^{x}} $
- $ {{x}^{n}}\,{{\log }_{a}}x.{{e}^{x}} $
The Correct Option is B
Solution and Explanation
$ ={{x}^{n}}.\frac{d}{dx}\{{{\log }_{a}}\,x.{{e}^{x}}\}+{{\log }_{a}}\,x.{{e}^{x}}\frac{d}{dx}({{x}^{n}}) $
$ ={{x}^{n}}\left\{ {{\log }_{a}}x.\frac{d}{dx}\,{{e}^{x}}+{{e}^{x}}\frac{d}{dx}{{\log }_{a}}x \right\} $
$ +{{e}^{x}}\,{{\log }_{a}}\,x.n{{x}^{n-1}} $
$ ={{x}^{n}}\left\{ {{e}^{x}}\,{{\log }_{a}}\,x+\frac{{{e}^{x}}}{x}.\frac{1}{{{\log }_{e}}a} \right\} $
$ +n{{x}^{n-1}}.{{e}^{x}}{{\log }_{a}}\,x $
$ ={{x}^{n-1}}\,{{e}^{x}}\left\{ x\,{{\log }_{a}}x+\frac{1}{{{\log }_{e}}\,a} \right\}+n{{x}^{n-1}}.{{e}^{x}}\,{{\log }_{a}}x $
$ ={{x}^{n-1}}\,.\,{{e}^{x}}\left\{ x\,{{\log }_{a}}x+\frac{1}{{{\log }_{e}}\,a}+n\,{{\log }_{a}}x \right\} $
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Concepts Used:
Exponential and Logarithmic Functions
Logarithmic Functions:
The inverses of exponential functions are the logarithmic functions. The exponential function is y = ax and its inverse is x = ay. The logarithmic function y = logax is derived as the equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, (where, a > 0, and a≠1). In totality, it is called the logarithmic function with base a.
The domain of a logarithmic function is real numbers greater than 0, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax w.r.t. the line y = x. This relationship is true for any of the exponential functions and their inverse.
Exponential Functions:
Exponential functions have the formation as:
f(x)=bx
where,
b = the base
x = the exponent (or power)