Calculus Examples

$y = 3 x^{3} (4 x^{4} - 7 x^{3})$

Step 1

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 $f (x) = 3 x^{3}$ and $g (x) = 4 x^{4} - 7 x^{3}$ .

$3 x^{3} \frac{d}{d x} [4 x^{4} - 7 x^{3}] + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 2

By the Sum Rule, the derivative of $4 x^{4} - 7 x^{3}$ with respect to $x$ is $\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 7 x^{3}]$ .

$3 x^{3} (\frac{d}{d x} [4 x^{4}] + \frac{d}{d x} [- 7 x^{3}]) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 3

Evaluate $\frac{d}{d x} [4 x^{4}]$ .

Tap for more steps...

Step 3.1

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

$3 x^{3} (4 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 7 x^{3}]) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$3 x^{3} (4 (4 x^{3}) + \frac{d}{d x} [- 7 x^{3}]) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 3.3

Multiply $4$ by $4$ .

$3 x^{3} (16 x^{3} + \frac{d}{d x} [- 7 x^{3}]) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 4

Evaluate $\frac{d}{d x} [- 7 x^{3}]$ .

Tap for more steps...

Step 4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{3}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{3}]$ .

$3 x^{3} (16 x^{3} - 7 \frac{d}{d x} [x^{3}]) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{3} (16 x^{3} - 7 (3 x^{2})) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 4.3

Multiply $3$ by $- 7$ .

$3 x^{3} (16 x^{3} - 21 x^{2}) + (4 x^{4} - 7 x^{3}) \frac{d}{d x} [3 x^{3}]$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$3 x^{3} (16 x^{3} - 21 x^{2}) + (4 x^{4} - 7 x^{3}) (3 \frac{d}{d x} [x^{3}])$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{3} (16 x^{3} - 21 x^{2}) + (4 x^{4} - 7 x^{3}) (3 (3 x^{2}))$

Step 5.3

Multiply $3$ by $3$ .

$3 x^{3} (16 x^{3} - 21 x^{2}) + (4 x^{4} - 7 x^{3}) (9 x^{2})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$3 x^{3} (16 x^{3}) + 3 x^{3} (- 21 x^{2}) + (4 x^{4} - 7 x^{3}) (9 x^{2})$

Step 6.2

Apply the distributive property.

$3 x^{3} (16 x^{3}) + 3 x^{3} (- 21 x^{2}) + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3

Combine terms.

Tap for more steps...

Step 6.3.1

Multiply $16$ by $3$ .

$48 x^{3} x^{3} + 3 x^{3} (- 21 x^{2}) + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.2

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 6.3.2.1

Move $x^{3}$ .

$48 (x^{3} x^{3}) + 3 x^{3} (- 21 x^{2}) + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.2.2

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

$48 x^{3 + 3} + 3 x^{3} (- 21 x^{2}) + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.2.3

Add $3$ and $3$ .

$48 x^{6} + 3 x^{3} (- 21 x^{2}) + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.3

Multiply $- 21$ by $3$ .

$48 x^{6} - 63 x^{3} x^{2} + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.4

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 6.3.4.1

Move $x^{2}$ .

$48 x^{6} - 63 (x^{2} x^{3}) + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.4.2

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

$48 x^{6} - 63 x^{2 + 3} + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.4.3

Add $2$ and $3$ .

$48 x^{6} - 63 x^{5} + 4 x^{4} (9 x^{2}) - 7 x^{3} (9 x^{2})$

Step 6.3.5

Multiply $9$ by $4$ .

$48 x^{6} - 63 x^{5} + 36 x^{4} x^{2} - 7 x^{3} (9 x^{2})$

Step 6.3.6

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

Tap for more steps...

Step 6.3.6.1

Move $x^{2}$ .

$48 x^{6} - 63 x^{5} + 36 (x^{2} x^{4}) - 7 x^{3} (9 x^{2})$

Step 6.3.6.2

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

$48 x^{6} - 63 x^{5} + 36 x^{2 + 4} - 7 x^{3} (9 x^{2})$

Step 6.3.6.3

Add $2$ and $4$ .

$48 x^{6} - 63 x^{5} + 36 x^{6} - 7 x^{3} (9 x^{2})$

Step 6.3.7

Multiply $9$ by $- 7$ .

$48 x^{6} - 63 x^{5} + 36 x^{6} - 63 x^{3} x^{2}$

Step 6.3.8

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 6.3.8.1

Move $x^{2}$ .

$48 x^{6} - 63 x^{5} + 36 x^{6} - 63 (x^{2} x^{3})$

Step 6.3.8.2

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

$48 x^{6} - 63 x^{5} + 36 x^{6} - 63 x^{2 + 3}$

Step 6.3.8.3

Add $2$ and $3$ .

$48 x^{6} - 63 x^{5} + 36 x^{6} - 63 x^{5}$

Step 6.3.9

Add $48 x^{6}$ and $36 x^{6}$ .

$84 x^{6} - 63 x^{5} - 63 x^{5}$

Step 6.3.10

Subtract $63 x^{5}$ from $- 63 x^{5}$ .

$84 x^{6} - 126 x^{5}$