Step by step partial derivative solution sample

Step by step solution of partial derivative, returned by our online calculator is given below:

Partial derivative calculator

Find partial derivative:

\(\frac{\partial^2}{\partial x \partial y}\left(x^3+y^3+2\cdot y \cdot x^2-8\cdot x \cdot y^2\right)\)

Input interpretation

Calculate partial derivative:

\(\frac{\partial^2}{\partial x \partial y}\left(x^3+y^3+2\cdot y \cdot x^2-8\cdot x \cdot y^2\right)\)

Answer:

\(4\cdot x-16\cdot y\)

Step by

Step 1

Calculate derivative of function \(x^3+y^3+2\cdot y \cdot x^2-8\cdot x \cdot y^2\) with respect to \(x\) variable:

\(\frac{\partial }{\partial y}\left(\frac{\partial }{\partial x}\left(x^3+y^3+2\cdot y \cdot x^2-8\cdot x \cdot y^2\right)\right)\)

Step 2

Differentiate the sum term by term:

\(\frac{\partial }{\partial y}\left(\frac{\partial x^3}{\partial x}+\frac{\partial y^3}{\partial x}+\frac{\partial }{\partial x}\left(2\cdot y \cdot x^2\right)-\frac{\partial }{\partial x}\left(8\cdot x \cdot y^2\right)\right)\)

Step 3

By using derivative identities \(\frac{\partial x^3}{\partial x}=3\cdot x^2\) , we get:

\(\frac{\partial }{\partial y}\left(3\cdot x^2+\frac{\partial y^3}{\partial x}+\frac{\partial }{\partial x}\left(2\cdot y \cdot x^2\right)-\frac{\partial }{\partial x}\left(8\cdot x \cdot y^2\right)\right)\)

Step 4

Derivative of constant expression \(\frac{\partial y^3}{\partial x}\) equals to zero:

\(\frac{\partial }{\partial y}\left(3\cdot x^2+\frac{\partial }{\partial x}\left(2\cdot y \cdot x^2\right)-\frac{\partial }{\partial x}\left(8\cdot x \cdot y^2\right)\right)\)

Step 5

Factor out constants:

\(\frac{\partial }{\partial y}\left(3\cdot x^2+2\cdot y \cdot \frac{\partial x^2}{\partial x}-8\cdot y^2 \cdot \frac{\partial x}{\partial x}\right)\)

Step 6

By using derivative identities \(\frac{\partial x^2}{\partial x}=2\cdot x\) , we get:

\(\frac{\partial }{\partial y}\left(3\cdot x^2+2\cdot y \cdot 2 \cdot x-8\cdot y^2 \cdot \frac{\partial x}{\partial x}\right)\)

Step 7

Derivative of \(x\) by variable \(x\) equals to \(1\):

\(\frac{\partial }{\partial y}\left(3\cdot x^2+2\cdot y \cdot 2 \cdot x-8\cdot y^2 \cdot 1\right)\)

Step 8

Differentiate the sum term by term:

\(\frac{\partial }{\partial y}\left(3\cdot x^2\right)+\frac{\partial }{\partial y}\left(4\cdot y \cdot x\right)-\frac{\partial }{\partial y}\left(8\cdot y^2\right)\)

Step 9

Derivative of constant expression \(\frac{\partial }{\partial y}\left(3\cdot x^2\right)\) equals to zero:

\(\frac{\partial }{\partial y}\left(4\cdot y \cdot x\right)-\frac{\partial }{\partial y}\left(8\cdot y^2\right)\)

Step 10

Factor out constants:

\(4\cdot x\cdot\frac{\partial y}{\partial y}-8\cdot\frac{\partial y^2}{\partial y}\)

Step 11

Derivative of \(y\) by variable \(y\) equals to \(1\):

\(4\cdot x-8\cdot\frac{\partial y^2}{\partial y}\)

Step 12

By using derivative identities \(\frac{\partial y^2}{\partial y}=2\cdot y\) , we get:

\(4\cdot x-16\cdot y\)

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Step by step partial derivative solution sample

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