Calculus Examples

$y = 6 x^{2} (4 x^{5} - 7 x^{2})$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2} (4 x^{5} - 7 x^{2})$ with respect to $x$ is $6 \frac{d}{d x} [x^{2} (4 x^{5} - 7 x^{2})]$ .

$6 \frac{d}{d x} [x^{2} (4 x^{5} - 7 x^{2})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = 4 x^{5} - 7 x^{2}$ .

$6 (x^{2} \frac{d}{d x} [4 x^{5} - 7 x^{2}] + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $4 x^{5} - 7 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [- 7 x^{2}]$ .

$6 (x^{2} (\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [- 7 x^{2}]) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{5}$ with respect to $x$ is $4 \frac{d}{d x} [x^{5}]$ .

$6 (x^{2} (4 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 7 x^{2}]) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$6 (x^{2} (4 (5 x^{4}) + \frac{d}{d x} [- 7 x^{2}]) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.4

Multiply $5$ by $4$ .

$6 (x^{2} (20 x^{4} + \frac{d}{d x} [- 7 x^{2}]) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{2}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{2}]$ .

$6 (x^{2} (20 x^{4} - 7 \frac{d}{d x} [x^{2}]) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$6 (x^{2} (20 x^{4} - 7 (2 x)) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.7

Multiply $2$ by $- 7$ .

$6 (x^{2} (20 x^{4} - 14 x) + (4 x^{5} - 7 x^{2}) \frac{d}{d x} [x^{2}])$

Step 3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$6 (x^{2} (20 x^{4} - 14 x) + (4 x^{5} - 7 x^{2}) (2 x))$

Step 3.9

Move $2$ to the left of $4 x^{5} - 7 x^{2}$ .

$6 (x^{2} (20 x^{4} - 14 x) + 2 \cdot (4 x^{5} - 7 x^{2}) x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$6 (x^{2} (20 x^{4}) + x^{2} (- 14 x) + 2 (4 x^{5} - 7 x^{2}) x)$

Step 4.2

Apply the distributive property.

$6 (x^{2} (20 x^{4}) + x^{2} (- 14 x) + (2 (4 x^{5}) + 2 (- 7 x^{2})) x)$

Step 4.3

Apply the distributive property.

$6 (x^{2} (20 x^{4}) + x^{2} (- 14 x) + 2 (4 x^{5}) x + 2 (- 7 x^{2}) x)$

Step 4.4

Apply the distributive property.

$6 (x^{2} (20 x^{4})) + 6 (x^{2} (- 14 x)) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5

Combine terms.

Tap for more steps...

Step 4.5.1

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 4.5.1.1

Move $x^{4}$ .

$6 (x^{4} x^{2} \cdot 20) + 6 (x^{2} (- 14 x)) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 (x^{4 + 2} \cdot 20) + 6 (x^{2} (- 14 x)) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.1.3

Add $4$ and $2$ .

$6 (x^{6} \cdot 20) + 6 (x^{2} (- 14 x)) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.2

Move $20$ to the left of $x^{6}$ .

$6 (20 \cdot x^{6}) + 6 (x^{2} (- 14 x)) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.3

Multiply $20$ by $6$ .

$120 x^{6} + 6 (x^{2} (- 14 x)) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.4

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 4.5.4.1

Move $x$ .

$120 x^{6} + 6 (x \cdot x^{2} \cdot - 14) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.4.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 4.5.4.2.1

Raise $x$ to the power of $1$ .

$120 x^{6} + 6 (x^{1} x^{2} \cdot - 14) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$120 x^{6} + 6 (x^{1 + 2} \cdot - 14) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.4.3

Add $1$ and $2$ .

$120 x^{6} + 6 (x^{3} \cdot - 14) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.5

Move $- 14$ to the left of $x^{3}$ .

$120 x^{6} + 6 (- 14 \cdot x^{3}) + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.6

Multiply $- 14$ by $6$ .

$120 x^{6} - 84 x^{3} + 6 (2 (4 x^{5}) x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.7

Multiply $4$ by $2$ .

$120 x^{6} - 84 x^{3} + 6 (8 x^{5} x) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.8

Raise $x$ to the power of $1$ .

$120 x^{6} - 84 x^{3} + 6 (8 (x^{1} x^{5})) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$120 x^{6} - 84 x^{3} + 6 (8 x^{1 + 5}) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.10

Add $1$ and $5$ .

$120 x^{6} - 84 x^{3} + 6 (8 x^{6}) + 6 (2 (- 7 x^{2}) x)$

Step 4.5.11

Multiply $8$ by $6$ .

$120 x^{6} - 84 x^{3} + 48 x^{6} + 6 (2 (- 7 x^{2}) x)$

Step 4.5.12

Multiply $- 7$ by $2$ .

$120 x^{6} - 84 x^{3} + 48 x^{6} + 6 (- 14 x^{2} x)$

Step 4.5.13

Raise $x$ to the power of $1$ .

$120 x^{6} - 84 x^{3} + 48 x^{6} + 6 (- 14 (x^{1} x^{2}))$

Step 4.5.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$120 x^{6} - 84 x^{3} + 48 x^{6} + 6 (- 14 x^{1 + 2})$

Step 4.5.15

Add $1$ and $2$ .

$120 x^{6} - 84 x^{3} + 48 x^{6} + 6 (- 14 x^{3})$

Step 4.5.16

Multiply $- 14$ by $6$ .

$120 x^{6} - 84 x^{3} + 48 x^{6} - 84 x^{3}$

Step 4.5.17

Add $120 x^{6}$ and $48 x^{6}$ .

$168 x^{6} - 84 x^{3} - 84 x^{3}$

Step 4.5.18

Subtract $84 x^{3}$ from $- 84 x^{3}$ .

$168 x^{6} - 168 x^{3}$