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  • Explain solution RD Sharma class 12 chapter 10 Differentiation exercise Very short answers question 16 maths

Explain solution RD Sharma class 12 chapter 10 Differentiation exercise Very short answers question 16 maths

Answers (1)

Answer:

The value of  \frac{d y}{d x} \text { at } x=e \text { is } 2 e^{e}

Hint:

Taking logarithm on both sides.

Given:

y=x^{x}

Solution:  

y=x^{x}

Taking logarithm on both sides.

\log y=x \log x

Differentiating with respect to x on both sides.,

\frac{1}{y} \frac{d y}{d x}=x \cdot \frac{1}{x}+1 . \log x

          \begin{aligned} &=1+\log x \\\\ &\Rightarrow \frac{d y}{d x}=y(1+\log x) \\\\ &\Rightarrow x^{x}(1+\log x) \end{aligned}

So at x=e,

\begin{aligned} &\frac{d y}{d x}=e^{e}(1+\log e) \\\\ &=e^{e}(1+1) \\\\ &=2 e^{e} \end{aligned}

 

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