Solve (3x)(5x)(3)+(3x)(2x)(3x)(2x)(2)-(11x)(2)+(3)

$$(3x)(5x)(3)+(3x)(2x)(3x)(2x)(2)-(11x)(2)+(3)$$

Evaluate

$$72x^{4}+45x^{2}-22x+3$$

Solution Steps

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

$$3x^{2}\times 5\times 3+3x\times 2x\times 3x\times 2x\times 2-11x\times 2+3$$

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

$$3x^{2}\times 5\times 3+3x^{2}\times 2\times 3x\times 2x\times 2-11x\times 2+3$$

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

$$3x^{2}\times 5\times 3+3x^{3}\times 2\times 3\times 2x\times 2-11x\times 2+3$$

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

$$3x^{2}\times 5\times 3+3x^{4}\times 2\times 3\times 2\times 2-11x\times 2+3$$

Multiply 3 and 5 to get 15.

$$15x^{2}\times 3+3x^{4}\times 2\times 3\times 2\times 2-11x\times 2+3$$

Multiply 15 and 3 to get 45.

$$45x^{2}+3x^{4}\times 2\times 3\times 2\times 2-11x\times 2+3$$

Multiply 3 and 2 to get 6.

$$45x^{2}+6x^{4}\times 3\times 2\times 2-11x\times 2+3$$

Multiply 6 and 3 to get 18.

$$45x^{2}+18x^{4}\times 2\times 2-11x\times 2+3$$

Multiply 18 and 2 to get 36.

$$45x^{2}+36x^{4}\times 2-11x\times 2+3$$

Multiply 36 and 2 to get 72.

$$45x^{2}+72x^{4}-11x\times 2+3$$

Multiply 11 and 2 to get 22.

$$45x^{2}+72x^{4}-22x+3$$

Differentiate w.r.t. x

$$288x^{3}+90x-22$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}\times 5\times 3+3x\times 2x\times 3x\times 2x\times 2-11x\times 2+3)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}\times 5\times 3+3x^{2}\times 2\times 3x\times 2x\times 2-11x\times 2+3)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}\times 5\times 3+3x^{3}\times 2\times 3\times 2x\times 2-11x\times 2+3)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}\times 5\times 3+3x^{4}\times 2\times 3\times 2\times 2-11x\times 2+3)$$

Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(15x^{2}\times 3+3x^{4}\times 2\times 3\times 2\times 2-11x\times 2+3)$$

Multiply 15 and 3 to get 45.

$$\frac{\mathrm{d}}{\mathrm{d}x}(45x^{2}+3x^{4}\times 2\times 3\times 2\times 2-11x\times 2+3)$$

Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(45x^{2}+6x^{4}\times 3\times 2\times 2-11x\times 2+3)$$

Multiply 6 and 3 to get 18.

$$\frac{\mathrm{d}}{\mathrm{d}x}(45x^{2}+18x^{4}\times 2\times 2-11x\times 2+3)$$

Multiply 18 and 2 to get 36.

$$\frac{\mathrm{d}}{\mathrm{d}x}(45x^{2}+36x^{4}\times 2-11x\times 2+3)$$

Multiply 36 and 2 to get 72.

$$\frac{\mathrm{d}}{\mathrm{d}x}(45x^{2}+72x^{4}-11x\times 2+3)$$

Multiply 11 and 2 to get 22.

$$\frac{\mathrm{d}}{\mathrm{d}x}(45x^{2}+72x^{4}-22x+3)$$

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}.

$$2\times 45x^{2-1}+4\times 72x^{4-1}-22x^{1-1}$$

Multiply 2 times 45.

$$90x^{2-1}+4\times 72x^{4-1}-22x^{1-1}$$

Subtract 1 from 2.

$$90x^{1}+4\times 72x^{4-1}-22x^{1-1}$$

Multiply 4 times 72.

$$90x^{1}+288x^{4-1}-22x^{1-1}$$

Subtract 1 from 4.

$$90x^{1}+288x^{3}-22x^{1-1}$$

Subtract 1 from 1.

$$90x^{1}+288x^{3}-22x^{0}$$

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

$$90x+288x^{3}-22x^{0}$$

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

$$90x+288x^{3}-22$$