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If a differentiable function $\mathrm{f}$ defined for $\mathrm{x}>0$ satisfies the relation $\mathrm{f}\left(\mathrm{x}^{2}\right)=\mathrm{x}^{3}, \mathrm{x}>0$, then what is the value of $\mathrm{f}^{\prime}(4) ?$
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3
According to given relation.
$\because \mathrm{f}\left(\mathrm{x}^{2}\right)=\mathrm{x}^{3}$
Putting $\mathrm{x}=\sqrt{\mathrm{x}}$
$\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{x}^{3 / 2}$
Differentiating both the sides,
$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\frac{3}{2} \mathrm{x}^{1 / 2}$
$\Rightarrow \quad f^{\prime}(4)=\frac{3}{2} \cdot 4^{1 / 2}=\frac{3}{2}(2)=3$
$\because \mathrm{f}\left(\mathrm{x}^{2}\right)=\mathrm{x}^{3}$
Putting $\mathrm{x}=\sqrt{\mathrm{x}}$
$\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{x}^{3 / 2}$
Differentiating both the sides,
$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=\frac{3}{2} \mathrm{x}^{1 / 2}$
$\Rightarrow \quad f^{\prime}(4)=\frac{3}{2} \cdot 4^{1 / 2}=\frac{3}{2}(2)=3$
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