Calculus Examples

${(2 x + y)}^{7}$

Step 1

Use the Binomial Theorem.

${(2 x)}^{7} + 7 {(2 x)}^{6} y + 21 {(2 x)}^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $2 x$ .

$2^{7} x^{7} + 7 {(2 x)}^{6} y + 21 {(2 x)}^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.2

Raise $2$ to the power of $7$ .

$128 x^{7} + 7 {(2 x)}^{6} y + 21 {(2 x)}^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.3

Apply the product rule to $2 x$ .

$128 x^{7} + 7 (2^{6} x^{6}) y + 21 {(2 x)}^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.4

Raise $2$ to the power of $6$ .

$128 x^{7} + 7 (64 x^{6}) y + 21 {(2 x)}^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.5

Multiply $64$ by $7$ .

$128 x^{7} + 448 x^{6} y + 21 {(2 x)}^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.6

Apply the product rule to $2 x$ .

$128 x^{7} + 448 x^{6} y + 21 (2^{5} x^{5}) y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.7

Raise $2$ to the power of $5$ .

$128 x^{7} + 448 x^{6} y + 21 (32 x^{5}) y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.8

Multiply $32$ by $21$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 35 {(2 x)}^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.9

Apply the product rule to $2 x$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 35 (2^{4} x^{4}) y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.10

Raise $2$ to the power of $4$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 35 (16 x^{4}) y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.11

Multiply $16$ by $35$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 35 {(2 x)}^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.12

Apply the product rule to $2 x$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 35 (2^{3} x^{3}) y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.13

Raise $2$ to the power of $3$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 35 (8 x^{3}) y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.14

Multiply $8$ by $35$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 280 x^{3} y^{4} + 21 {(2 x)}^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.15

Apply the product rule to $2 x$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 280 x^{3} y^{4} + 21 (2^{2} x^{2}) y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.16

Raise $2$ to the power of $2$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 280 x^{3} y^{4} + 21 (4 x^{2}) y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.17

Multiply $4$ by $21$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 280 x^{3} y^{4} + 84 x^{2} y^{5} + 7 (2 x) y^{6} + y^{7}$

Step 2.18

Multiply $2$ by $7$ .

$128 x^{7} + 448 x^{6} y + 672 x^{5} y^{2} + 560 x^{4} y^{3} + 280 x^{3} y^{4} + 84 x^{2} y^{5} + 14 x y^{6} + y^{7}$