Question

Find the four second partial derivatives of the following functions. h(x,y)=x^{3}+xy^{2}+1

Derivatives
ANSWERED
asked 2021-06-04
Find the four second partial derivatives of the following functions.
\(\displaystyle{h}{\left({x},{y}\right)}={x}^{{{3}}}+{x}{y}^{{{2}}}+{1}\)

Answers (1)

2021-06-05
Step 1
Given information:
\(\displaystyle{h}{\left({x},{y}\right)}={x}^{{{3}}}+{x}{y}^{{{2}}}+{1}\)
Step 2
First find the partial derivative of the given function,
\(\displaystyle{\frac{{\partial{h}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({x}^{{{3}}}+{x}{y}^{{{2}}}+{1}\right)}\)
\(\displaystyle={3}{x}^{{{2}}}+{y}^{{{2}}}+{0}\)
\(\displaystyle={3}{x}^{{{2}}}+{y}^{{{2}}}\)
Now find the required double partial derivative of given function,
\(\displaystyle{\frac{{\partial^{{{2}}}{h}}}{{\partial{x}^{{{2}}}}}}={\frac{{\partial}}{{\partial{x}}}}{\left({3}{x}^{{{2}}}+{y}^{{{2}}}\right)}\)
=6x+0+0
=6x
0
 
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