Partial Derivatives

Practice Questions

Calculus 3 › Partial Derivatives

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$\begin{align*}&\text{Calculate the limits of the function}\&\lim_{(x,y)\rightarrow(0+,0-)}\frac{(18xy^{3})}{(19x^{4} + 18y^{4})}\&\text{along the y-axis and the line, }y=x\end{align*}$

$\begin{align*}&\text{Calculate the limits of the function}\&\lim_{(x,y)\rightarrow(0-,0+)}-\frac{(2x^{2}y^{4} - 5x^{3}y^{3} + 3x^{5}y)}{(3x^{6} + 4y^{6})}\&\text{along the y-axis and the line, }y=x\end{align*}$

Given the function $f(x,y)=\sin x^2y$ , find the partial derivative .

$\begin{align*}&\text{Calculate the limits of the function}\&lim_{(x,y)\rightarrow(0+,0-)}-\frac{(4x^{2}y - 11xy^{2} + 7x^{3})}{(5x^{3} + 4y^{3})}\&\text{along the y-axis and the line, }y=x\end{align*}$

$\begin{align*}&\text{Perform the partial derivation, }f_{xy}\&\text{Where }f(x,y,z)=5x^{3}zsin{(y^{3})} - sin{(x^{2})}sin{(4y)}sin{(4z)}\end{align*}$

$\begin{align*}&\text{Calculate the limits of the function}\&lim_{(x,y)\rightarrow(0-,0-)}-\frac{(14xy^{3} + 12x^{4} + 16y^{4})}{(8x^{2}y^{2} + 19xy^{3} + 5x^{4} + y^{4})}\&\text{along the x-axis and the line, }y=x\end{align*}$

$\begin{align*}&\text{Evaluate the following limit:}\&lim_{x\rightarrow1-}\frac{-4\cdot (x - 1)^{2}}{9sin(2\cdot \pi x)^{2}}\end{align*}$

$\begin{align*}&\text{Find }D_{\overrightarrow{u}}(x,y,z)\text{ where }f(x,y,z)=-xsin{(z)}\&\text{In the direction of }\overrightarrow{v}=(-13,19,-10).\end{align*}$

$\begin{align*}&\text{Evaluate the following limit:}\&lim_{x\rightarrow8-}-\frac{(13200x + 7403x^{2} - 803x^{3} - 99x^{4} + 11x^{5} - 123200)}{(1456x^{2} - 13088x + 128x^{3} - 16x^{4} + 11520)}\end{align*}$

$\begin{align*}&\text{Calculate the limits of the function}\&\lim_{(x,y)\rightarrow(0-,0+)}\frac{(11xy^{5} + 14x^{6})}{(20x^{5}y - 5x^{6} + 5y^{6})}\&\text{along the y-axis and the line, }y=x\end{align*}$

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