CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Differentiate the following w.r.t. x: $\frac{8^\text{x}}{\text{x}^8}$
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Answer
Let $\text{y}=\frac{8^\text{x}}{\text{x}^8}$ $\Rightarrow\ \log\text{y}=\log\frac{8^\text{x}}{\text{x}^8}$ $\Rightarrow\ \frac{\text{d}}{\text{dy}}\log\text{y}\cdot\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big[\log8^\text{x}-\log\text{x}^8\big]$ $\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\big[\text{x}.\log8-8.\log\text{x}]$ On differentiating w.r.t. x, we get $\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\log8.1-8.\frac{1}{\text{x}}$ $\Rightarrow\ \frac{1}{\text{y}}\cdot\frac{\text{dy}}{\text{dx}}=\log8-\frac{8}{\text{x}}$ $\therefore\ \frac{\text{dy}}{\text{dx}}=\text{y}\Big(\log8-\frac{8}{\text{x}}\Big)=\frac{8^\text{x}}{\text{x}^8}\Big(\log8-\frac{8}{\text{x}}\Big)$
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