Solve (x^{4})(-x^{3})+(5xy)(xy^{2})+(-4x^{2})(-x^5)-(2x)(-x^4)+4-(-x^2)(-x^3)

$$( x ^ { 4 } ) ( - x ^ { 3 } ) + ( 5 x y ) ( x y ^ { 2 } ) + ( - 4 x ^ { 2 } ) ( - x ^ { 5 } ) - ( 2 x ) ( - x ^ { 4 } ) + 4 - ( - x ^ { 2 } ) ( - x ^ { 3 } )$$

Differentiate w.r.t. x

$$x\left(21x^{5}+5x^{3}+10y^{3}\right)$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}\left(-x^{3}\right)+5x^{2}yy^{2}-4x^{2}\left(-x^{5}\right)-2x\left(-x^{4}\right)+4-\left(-x^{2}\right)\left(-x^{3}\right))$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

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

Multiply -4 and -1 to get 4.

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

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{4}\left(-x^{3}\right)+5x^{2}y^{3}+4x^{7}-2x\left(-x^{4}\right)+4-\left(-x^{2}\right)\left(-x^{3}\right))$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{7}\left(-1\right)+5x^{2}y^{3}+4x^{7}-2x\left(-1\right)x^{4}+4-\left(-x^{2}\left(-1\right)x^{3}\right))$$

Combine x^{7}\left(-1\right) and 4x^{7} to get 3x^{7}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{7}+5x^{2}y^{3}-2x\left(-1\right)x^{4}+4-\left(-x^{2}\left(-1\right)x^{3}\right))$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{7}+5x^{2}y^{3}-2x^{5}\left(-1\right)+4-\left(-x^{2}\left(-1\right)x^{3}\right))$$

Multiply -2 and -1 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{7}+5x^{2}y^{3}+2x^{5}+4-\left(-x^{2}\left(-1\right)x^{3}\right))$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{7}+5x^{2}y^{3}+2x^{5}+4-\left(-x^{5}\left(-1\right)\right))$$

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{7}+5x^{2}y^{3}+2x^{5}+4+x^{5}\left(-1\right))$$

Combine 2x^{5} and x^{5}\left(-1\right) to get x^{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{7}+5x^{2}y^{3}+x^{5}+4)$$

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}.

$$7\times 3x^{7-1}+2\times 5y^{3}x^{2-1}+5x^{5-1}$$

Multiply 7 times 3.

$$21x^{7-1}+2\times 5y^{3}x^{2-1}+5x^{5-1}$$

Subtract 1 from 7.

$$21x^{6}+2\times 5y^{3}x^{2-1}+5x^{5-1}$$

Multiply 2 times 5y^{3}.

$$21x^{6}+10y^{3}x^{2-1}+5x^{5-1}$$

Subtract 1 from 2.

$$21x^{6}+10y^{3}x^{1}+5x^{5-1}$$

Subtract 1 from 5.

$$21x^{6}+10y^{3}x^{1}+5x^{4}$$

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

$$21x^{6}+10y^{3}x+5x^{4}$$

Evaluate

$$3x^{7}+x^{5}+5x^{2}y^{3}+4$$