Solve x^4y^2-(-5/4x^3y)(-1/5xy)+8x^3(-1/8x{y}^2)-3/2{x}^4{y}^2

$${ x }^{ 4 } { y }^{ 2 } -(- \frac{ 5 }{ 4 } { x }^{ 3 } y)(- \frac{ 1 }{ 5 } xy)+8 { x }^{ 3 } (- \frac{ 1 }{ 8 } x { y }^{ 2 } )- \frac{ 3 }{ 2 } { x }^{ 4 } { y }^{ 2 }$$

Evaluate

$$-\frac{7y^{2}x^{4}}{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$x^{4}y^{2}-\left(-\frac{5}{4}x^{4}y\left(-\frac{1}{5}\right)y\right)+8x^{3}\left(-\frac{1}{8}\right)xy^{2}-\frac{3}{2}x^{4}y^{2}$$

Multiply y and y to get y^{2}.

$$x^{4}y^{2}-\left(-\frac{5}{4}x^{4}y^{2}\left(-\frac{1}{5}\right)\right)+8x^{3}\left(-\frac{1}{8}\right)xy^{2}-\frac{3}{2}x^{4}y^{2}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$x^{4}y^{2}-\left(-\frac{5}{4}x^{4}y^{2}\left(-\frac{1}{5}\right)\right)+8x^{4}\left(-\frac{1}{8}\right)y^{2}-\frac{3}{2}x^{4}y^{2}$$

Multiply -\frac{5}{4} and -\frac{1}{5} to get \frac{1}{4}.

$$x^{4}y^{2}-\frac{1}{4}x^{4}y^{2}+8x^{4}\left(-\frac{1}{8}\right)y^{2}-\frac{3}{2}x^{4}y^{2}$$

Combine x^{4}y^{2} and -\frac{1}{4}x^{4}y^{2} to get \frac{3}{4}x^{4}y^{2}.

$$\frac{3}{4}x^{4}y^{2}+8x^{4}\left(-\frac{1}{8}\right)y^{2}-\frac{3}{2}x^{4}y^{2}$$

Multiply 8 and -\frac{1}{8} to get -1.

$$\frac{3}{4}x^{4}y^{2}-x^{4}y^{2}-\frac{3}{2}x^{4}y^{2}$$

Combine \frac{3}{4}x^{4}y^{2} and -x^{4}y^{2} to get -\frac{1}{4}x^{4}y^{2}.

$$-\frac{1}{4}x^{4}y^{2}-\frac{3}{2}x^{4}y^{2}$$

Combine -\frac{1}{4}x^{4}y^{2} and -\frac{3}{2}x^{4}y^{2} to get -\frac{7}{4}x^{4}y^{2}.

$$-\frac{7}{4}x^{4}y^{2}$$

Differentiate w.r.t. x

$$-7y^{2}x^{3}$$