Solve -5x^2y+7xy^2-2x^3y+8

$$- 5 x ^ { 2 } y + 7 x y ^ { 2 } - 2 x ^ { 3 } y + 8$$

Differentiate w.r.t. x

$$y\left(7y-10x-6x^{2}\right)$$

Steps Using Derivative Rule for Sum

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$2\left(-5y\right)x^{2-1}+7y^{2}x^{1-1}+3\left(-2y\right)x^{3-1}$$

Multiply 2 times -5y.

$$\left(-10y\right)x^{2-1}+7y^{2}x^{1-1}+3\left(-2y\right)x^{3-1}$$

Subtract 1 from 2.

$$\left(-10y\right)x^{1}+7y^{2}x^{1-1}+3\left(-2y\right)x^{3-1}$$

Subtract 1 from 1.

$$\left(-10y\right)x^{1}+7y^{2}x^{0}+3\left(-2y\right)x^{3-1}$$

Multiply 1 times 7y^{2}.

$$\left(-10y\right)x^{1}+7y^{2}x^{0}+\left(-6y\right)x^{3-1}$$

Subtract 1 from 3.

$$\left(-10y\right)x^{1}+7y^{2}x^{0}+\left(-6y\right)x^{2}$$

For any term t, t^{1}=t.

$$\left(-10y\right)x+7y^{2}x^{0}+\left(-6y\right)x^{2}$$

For any term t except 0, t^{0}=1.

$$\left(-10y\right)x+7y^{2}\times 1+\left(-6y\right)x^{2}$$

For any term t, t\times 1=t and 1t=t.

$$\left(-10y\right)x+7y^{2}+\left(-6y\right)x^{2}$$

Evaluate

$$7xy^{2}-2yx^{3}-5yx^{2}+8$$