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  • Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 19

Need solution for RD Sharma maths class 12 chapter Differentiation exercise 10.5 question 19

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Answer: \frac{d y}{d x}=e^{x}+10^{x} \log 10+x^{x}(\log n+1)

Hint:  Differentiate the statement taking u and v

Given: y=e^{x}+10^{x}+x^{x}

Solution:  
Let x^{x}=u

        \begin{aligned} &y=e^{x}+10^{x}+u \\\\ &\frac{d y}{d x}=\frac{d\left(e^{x}\right)}{d x}+\frac{d\left(10^{x}\right)}{d x}+\frac{d u}{d x} \end{aligned}        ..........(1)

For \mathrm{u}, u=x^{x}

        \begin{aligned} &\log u=\log x^{x} \\\\ &\frac{d}{d x}(\log u)=x \log x \end{aligned}

        \frac{1}{u} \frac{d u}{d x}=\log x+x \cdot \frac{d}{d x}(\log x)

        \begin{aligned} &\frac{d u}{d x}=u\left[\log x+\not x \cdot \frac{1}{\not{x}}\right] \\\\ &=x^{x}(\log x+1) \end{aligned}

Put \frac{d u}{d x}  in eq (1)

        \begin{aligned} &\frac{d}{d x}\left(e^{x}\right)=e^{x} \\\\ &\frac{d y}{d x}=e^{x}+10^{x} \log 10+x^{x}(\log x+1) \end{aligned}

 

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