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$$
\text { If } f(x)=\alpha x^n, \text { prove that } \alpha=\frac{f^{\prime}(1)}{n}
$$
\text { If } f(x)=\alpha x^n, \text { prove that } \alpha=\frac{f^{\prime}(1)}{n}
$$
Solution:
2813 Upvotes
Verified Answer
$f(x)=\alpha x^n$
$$
\begin{aligned}
&\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\alpha \cdot \mathrm{nx}^{\mathrm{n}-1} \Rightarrow \mathrm{f}^{\prime}(1)=\alpha \mathrm{n} \cdot(1)^{\mathrm{n}-1}=\alpha \mathrm{n} \\
&\Rightarrow \alpha=\frac{\mathrm{f}^{\prime}(\mathrm{l})}{\mathrm{n}}
\end{aligned}
$$
$$
\begin{aligned}
&\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\alpha \cdot \mathrm{nx}^{\mathrm{n}-1} \Rightarrow \mathrm{f}^{\prime}(1)=\alpha \mathrm{n} \cdot(1)^{\mathrm{n}-1}=\alpha \mathrm{n} \\
&\Rightarrow \alpha=\frac{\mathrm{f}^{\prime}(\mathrm{l})}{\mathrm{n}}
\end{aligned}
$$
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