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Question
f\left(x,y\right)=x^{3}+3xy
Function
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\text{Find the first partial derivative with respect to }x
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\text{Find the first partial derivative with respect to }y
f_{x}=3x^{2}+3y
Simplify
f\left(x,y\right)=x^{3}+3xy
\text{Find the first partial derivative by treating the variable }y\text{ as a constant and differentiating with respect to }x
f_{x}=\frac{\partial}{\partial x}\left(x^{3}+3xy\right)
\text{Use differentiation rule }\frac{\partial}{\partial x}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{\partial}{\partial x}\left(f\left(x\right)\right)\pm \frac{\partial}{\partial x}(g(x))
f_{x}=\frac{\partial}{\partial x}\left(x^{3}\right)+\frac{\partial}{\partial x}\left(3xy\right)
\text{Use }\frac{\partial}{\partial x} x^{n}=n x^{n-1}\text{ to find derivative}
f_{x}=3x^{2}+\frac{\partial}{\partial x}\left(3xy\right)
Solution
More Steps
Evaluate
\frac{\partial}{\partial x}\left(3xy\right)
\text{Use differentiation rule }\frac{\partial}{\partial x}\left(cf\left(x\right)\right)=c\times\frac{\partial}{\partial x}(f(x))
3y\times \frac{\partial}{\partial x}\left(x\right)
\text{Use }\frac{\partial}{\partial x} x^{n}=n x^{n-1}\text{ to find derivative}
3y\times 1
Multiply the terms
3y
f_{x}=3x^{2}+3y
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