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  • Differentiate each of the following w.r.t. x 8^x/x^8

Differentiate each of the following w.r.t. x

\frac{8^x}{x^8}

Answers (1)

We have been given  \frac{8^x}{x^8}

Let us Assume y=\frac{8^x}{x^8}

Now, taking log on both sides, we get

\log y=\log \frac{8^{x}}{x^{8}}
since we know,  \log \frac{\mathrm{m}}{\mathrm{n}}=\log \mathrm{m}-\log \mathrm{n}
\log y=\log 8^{x}-\log x^{8}
since we know,  \log \mathrm{a}^{n}=n \log a
\log y=x \log 8-8 \log {x}
Now, Differentiate w.r.t x
\\\frac{1}{y} \frac{d y}{d x}=\frac{d}{d x}(x \log 8-8 \log x)$ \\$\frac{d y}{d x}=y\left(1 \times \log 8-8 \times \frac{1}{x}\right)

\frac{d y}{d x}=\frac{8^{x}}{x^{8}}\left(\log 8-\frac{8}{x}\right)
Hence, \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{8^{\mathrm{x}}}{\mathrm{x}^{8}}\left(\log 8-\frac{8}{\mathrm{x}}\right)

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infoexpert22

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