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Question

# In given problem, find all first partial derivatives of each function.$f(x, y)=\left(4 x-y^2\right)^{3 / 2}$

Solution

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Given

$f\left( x,y\right)=\left(4x-y^{2} \right)^{\frac{3}{2}}$

We treat $y$ as a constant and differentiable with respect to $x$,obtaining;

\begin{align*} f_{x}\left(x,y \right)&=\dfrac{\partial f\left(x,y \right)}{\partial x}\\ &=\dfrac{\partial }{\partial x}\left( 4x-y^{2}\right)^{\frac{3}{2}}\\ &=\dfrac{3}{2}\cdot4\left( 4x-y^{2}\right)^{\frac{3}{2}-1}\\ &=6\left(4x-y^{2} \right)^{\frac{1}{2}}\\ &=6\sqrt{4x-y^{2}}\end{align*}

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