Question
Differentiate the following function with respect to x:
$\text{x}^5\text{e}^\text{x}+\text{x}^6\log\text{x}$
$\text{x}^5\text{e}^\text{x}+\text{x}^6\log\text{x}$
$=\text{x}^5\frac{\text{d}}{\text{dx}}(\text{e}^\text{x})+\text{e}^\text{x}\frac{\text{d}}{\text{dx}}(\text{x}^5)+(\text{x})^6\frac{\text{d}}{\text{dx}}(\log\text{x})+\log\text{x}\frac{\text{d}}{\text{dx}}(\text{x}^6)$
$=\text{x}^5\text{e}^\text{x}+\text{e}^\text{x}(5\text{x}^4)+\text{x}^6.\frac{1}{\text{x}}+\log\text{x}(6\text{x}^5)$
$=\text{x}^5\text{e}^\text{x}+\text{e}^\text{x}(5\text{x}^4)+\text{x}^5+\log\text{x}(6\text{x}^5)$
$=\text{x}^4(\text{x}\text{e}^\text{x}+5\text{e}^\text{x}+\text{x}+\text{6x}+\log\text{x})$
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The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.