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Find the partial derivative of the dependent variable or function with respect to each of the independent variables.$$z=\left(3 x^{2}+x y^{3}\right)^{4}$$

$\frac{\partial z}{\partial y}=12 x y^{2}\left(3 x^{2}+x y^{3}\right)^{3}$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 3

Partial Derivatives

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Okay we want to find the first partial derivatives of the function I have X. Y. Is he going three X squared plus x Y cube to the power of four. This question is challenging our understanding of differentiation and applied to our newfound multi variant functions. So the first partial divers are searching for our DF DX and DY where for each we have to use standard differentiation techniques. We learned in single very real calculus. But for DF dx, n E Y term street is a constant and for the FBI any extreme street is a constant. So for fx we have to start off with a chain role. So a derivative ist four times three X squared plus X Y cube cubes 10 6 X plus Y cube. Where six X Y cube is different is a derivative of what's inside parentheses. Similarly, F Y is four times three X squared plus X Y cube cube time, zero plus three X Y squared. The reason there is now a zero is because the three X squared and fantasies does not have a term of why they're what's constantly goes to zero. Thus fy is 12 X Y squared term three X squared plus x Y squared, cute.

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