Evaluate the derivative of 8x-x8

Let y=8^x/x^8⇒ log y=log (8^x/x^8) ⇒d/dy(log y).dy/dx=d/dxlog[8^x logx^8] ⇒1/y.dy/dx=d/dx[x.log 8 8.log x] On diffrentiating w.r.t x, we get 1/y.dy/dx=log 8.1 ...

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Standard XII

Mathematics

Higher Order Derivatives

Evaluate the ...

Question

Evaluate the derivative of $\frac{8^{x}}{x^{8}}$

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Solution

Let $y = \frac{8^{x}}{x^{8}} \Rightarrow l o g y = l o g (\frac{8^{x}}{x^{8}})$
$\Rightarrow \frac{d}{d y} (l o g y) . \frac{d y}{d x} = \frac{d}{d x} l o g [8^{x} - l o g x^{8}] \Rightarrow \frac{1}{y} . \frac{d y}{d x} = \frac{d}{d x} [x . l o g 8 - 8. l o g x]$

On diffrentiating w.r.t x, we get
$\frac{1}{y} . \frac{d y}{d x} = l o g 8.1 - 8. \frac{1}{x} \Rightarrow \frac{1}{y} . \frac{d y}{d x} = l o g 8 - \frac{8}{x} ∴ \frac{d y}{d x} = y (l o g 8 - \frac{8}{x}) = \frac{8^{x}}{x^{8}} (l o g 8 - \frac{8}{x})$

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Evaluate the derivative of 8xx8

Let y=8xx8⇒log y=log(8xx8) ⇒ddy(log y).dydx=ddxlog[8x−logx8]⇒1y.dydx=ddx[x.log 8−8.log x] On diffrentiating w.r.t x, we get 1y.dydx=log 8.1−8.1x⇒1y.dydx=log 8−8x∴ dydx=y(log 8−8x)=8xx8(log 8−8x)

Evaluate the derivative of $\frac{8^{x}}{x^{8}}$