Solve (1+x)(1-x)(1+x^2x(+x^4)

$$( 1 + x ) ( 1 - x ) ( 1 + x ^ { 2 } x ( + x ^ { 4 } )$$

Evaluate

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

Solution Steps

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

$$\left(1+x\right)\left(1-x\right)\left(1+x^{3}x^{4}\right)$$

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

$$\left(1+x\right)\left(1-x\right)\left(1+x^{7}\right)$$

Use the distributive property to multiply 1+x by 1-x and combine like terms.

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

Use the distributive property to multiply 1-x^{2} by 1+x^{7}.

$$1+x^{7}-x^{2}-x^{9}$$

Differentiate w.r.t. x

$$x\left(-9x^{7}+7x^{5}-2\right)$$

Steps Using Derivative Rule for Sum

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(1+x\right)\left(1-x\right)\left(1+x^{7}\right))$$

Use the distributive property to multiply 1+x by 1-x and combine like terms.

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

Use the distributive property to multiply 1-x^{2} by 1+x^{7}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(1+x^{7}-x^{2}-x^{9})$$

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}.

$$7x^{7-1}+2\left(-1\right)x^{2-1}+9\left(-1\right)x^{9-1}$$

Subtract 1 from 7.

$$7x^{6}+2\left(-1\right)x^{2-1}+9\left(-1\right)x^{9-1}$$

Multiply 2 times -1.

$$7x^{6}-2x^{2-1}+9\left(-1\right)x^{9-1}$$

Subtract 1 from 2.

$$7x^{6}-2x^{1}+9\left(-1\right)x^{9-1}$$

Multiply 2 times -1.

$$7x^{6}-2x^{1}-9x^{9-1}$$

Subtract 1 from 9.

$$7x^{6}-2x^{1}-9x^{8}$$

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

$$7x^{6}-2x-9x^{8}$$