Calculus Examples

$y = 3 x^{5} {(3 x - 4)}^{2}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Rewrite ${(3 x - 4)}^{2}$ as $(3 x - 4) (3 x - 4)$ .

$\frac{d}{d x} [3 x^{5} ((3 x - 4) (3 x - 4))]$

Step 1.2

Expand $(3 x - 4) (3 x - 4)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\frac{d}{d x} [3 x^{5} (3 x (3 x - 4) - 4 (3 x - 4))]$

Step 1.2.2

Apply the distributive property.

$\frac{d}{d x} [3 x^{5} (3 x (3 x) + 3 x \cdot - 4 - 4 (3 x - 4))]$

Step 1.2.3

Apply the distributive property.

$\frac{d}{d x} [3 x^{5} (3 x (3 x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4)]$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [3 x^{5} (3 \cdot 3 x \cdot x + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4)]$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.3.1.2.1

Move $x$ .

$\frac{d}{d x} [3 x^{5} (3 \cdot 3 (x \cdot x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4)]$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [3 x^{5} (3 \cdot 3 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4)]$

Step 1.3.1.3

Multiply $3$ by $3$ .

$\frac{d}{d x} [3 x^{5} (9 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4)]$

Step 1.3.1.4

Multiply $- 4$ by $3$ .

$\frac{d}{d x} [3 x^{5} (9 x^{2} - 12 x - 4 (3 x) - 4 \cdot - 4)]$

Step 1.3.1.5

Multiply $3$ by $- 4$ .

$\frac{d}{d x} [3 x^{5} (9 x^{2} - 12 x - 12 x - 4 \cdot - 4)]$

Step 1.3.1.6

Multiply $- 4$ by $- 4$ .

$\frac{d}{d x} [3 x^{5} (9 x^{2} - 12 x - 12 x + 16)]$

Step 1.3.2

Subtract $12 x$ from $- 12 x$ .

$\frac{d}{d x} [3 x^{5} (9 x^{2} - 24 x + 16)]$

Step 1.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5} (9 x^{2} - 24 x + 16)$ with respect to $x$ is $3 \frac{d}{d x} [x^{5} (9 x^{2} - 24 x + 16)]$ .

$3 \frac{d}{d x} [x^{5} (9 x^{2} - 24 x + 16)]$

Step 1.5

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 $f (x) = x^{5}$ and $g (x) = 9 x^{2} - 24 x + 16$ .

$3 (x^{5} \frac{d}{d x} [9 x^{2} - 24 x + 16] + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6

Differentiate.

Tap for more steps...

Step 1.6.1

By the Sum Rule, the derivative of $9 x^{2} - 24 x + 16$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [16]$ .

$3 (x^{5} (\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$3 (x^{5} (9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.3

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

$3 (x^{5} (9 (2 x) + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.4

Multiply $2$ by $9$ .

$3 (x^{5} (18 x + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.5

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$3 (x^{5} (18 x - 24 \frac{d}{d x} [x] + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.6

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

$3 (x^{5} (18 x - 24 \cdot 1 + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.7

Multiply $- 24$ by $1$ .

$3 (x^{5} (18 x - 24 + \frac{d}{d x} [16]) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.8

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$3 (x^{5} (18 x - 24 + 0) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.9

Add $18 x - 24$ and $0$ .

$3 (x^{5} (18 x - 24) + (9 x^{2} - 24 x + 16) \frac{d}{d x} [x^{5}])$

Step 1.6.10

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

$3 (x^{5} (18 x - 24) + (9 x^{2} - 24 x + 16) (5 x^{4}))$

Step 1.6.11

Move $5$ to the left of $9 x^{2} - 24 x + 16$ .

$3 (x^{5} (18 x - 24) + 5 \cdot (9 x^{2} - 24 x + 16) x^{4})$

Step 1.7

Simplify.

Tap for more steps...

Step 1.7.1

Apply the distributive property.

$3 (x^{5} (18 x) + x^{5} \cdot - 24 + 5 (9 x^{2} - 24 x + 16) x^{4})$

Step 1.7.2

Apply the distributive property.

$3 (x^{5} (18 x) + x^{5} \cdot - 24 + (5 (9 x^{2}) + 5 (- 24 x) + 5 \cdot 16) x^{4})$

Step 1.7.3

Apply the distributive property.

$3 (x^{5} (18 x) + x^{5} \cdot - 24 + 5 (9 x^{2}) x^{4} + 5 (- 24 x) x^{4} + 5 \cdot 16 x^{4})$

Step 1.7.4

Apply the distributive property.

$3 (x^{5} (18 x)) + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5

Combine terms.

Tap for more steps...

Step 1.7.5.1

Multiply $x^{5}$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.7.5.1.1

Move $x$ .

$3 (x \cdot x^{5} \cdot 18) + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.1.2

Multiply $x$ by $x^{5}$ .

Tap for more steps...

Step 1.7.5.1.2.1

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

$3 (x^{1} x^{5} \cdot 18) + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.1.2.2

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

$3 (x^{1 + 5} \cdot 18) + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.1.3

Add $1$ and $5$ .

$3 (x^{6} \cdot 18) + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.2

Move $18$ to the left of $x^{6}$ .

$3 (18 \cdot x^{6}) + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.3

Multiply $18$ by $3$ .

$54 x^{6} + 3 (x^{5} \cdot - 24) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.4

Move $- 24$ to the left of $x^{5}$ .

$54 x^{6} + 3 (- 24 \cdot x^{5}) + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.5

Multiply $- 24$ by $3$ .

$54 x^{6} - 72 x^{5} + 3 (5 (9 x^{2}) x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.6

Multiply $9$ by $5$ .

$54 x^{6} - 72 x^{5} + 3 (45 x^{2} x^{4}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.7

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

Tap for more steps...

Step 1.7.5.7.1

Move $x^{4}$ .

$54 x^{6} - 72 x^{5} + 3 (45 (x^{4} x^{2})) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.7.2

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

$54 x^{6} - 72 x^{5} + 3 (45 x^{4 + 2}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.7.3

Add $4$ and $2$ .

$54 x^{6} - 72 x^{5} + 3 (45 x^{6}) + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.8

Multiply $45$ by $3$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} + 3 (5 (- 24 x) x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.9

Multiply $- 24$ by $5$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} + 3 (- 120 x \cdot x^{4}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.10

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

Tap for more steps...

Step 1.7.5.10.1

Move $x^{4}$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} + 3 (- 120 (x^{4} x)) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.10.2

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 1.7.5.10.2.1

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

$54 x^{6} - 72 x^{5} + 135 x^{6} + 3 (- 120 (x^{4} x^{1})) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.10.2.2

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

$54 x^{6} - 72 x^{5} + 135 x^{6} + 3 (- 120 x^{4 + 1}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.10.3

Add $4$ and $1$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} + 3 (- 120 x^{5}) + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.11

Multiply $- 120$ by $3$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} - 360 x^{5} + 3 (5 \cdot 16 x^{4})$

Step 1.7.5.12

Multiply $5$ by $16$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} - 360 x^{5} + 3 (80 x^{4})$

Step 1.7.5.13

Multiply $80$ by $3$ .

$54 x^{6} - 72 x^{5} + 135 x^{6} - 360 x^{5} + 240 x^{4}$

Step 1.7.5.14

Add $54 x^{6}$ and $135 x^{6}$ .

$189 x^{6} - 72 x^{5} - 360 x^{5} + 240 x^{4}$

Step 1.7.5.15

Subtract $360 x^{5}$ from $- 72 x^{5}$ .

$f' (x) = 189 x^{6} - 432 x^{5} + 240 x^{4}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $189 x^{6} - 432 x^{5} + 240 x^{4}$ with respect to $x$ is $\frac{d}{d x} [189 x^{6}] + \frac{d}{d x} [- 432 x^{5}] + \frac{d}{d x} [240 x^{4}]$ .

$\frac{d}{d x} [189 x^{6}] + \frac{d}{d x} [- 432 x^{5}] + \frac{d}{d x} [240 x^{4}]$

Step 2.2

Evaluate $\frac{d}{d x} [189 x^{6}]$ .

Tap for more steps...

Step 2.2.1

Since $189$ is constant with respect to $x$ , the derivative of $189 x^{6}$ with respect to $x$ is $189 \frac{d}{d x} [x^{6}]$ .

$189 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 432 x^{5}] + \frac{d}{d x} [240 x^{4}]$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$189 (6 x^{5}) + \frac{d}{d x} [- 432 x^{5}] + \frac{d}{d x} [240 x^{4}]$

Step 2.2.3

Multiply $6$ by $189$ .

$1134 x^{5} + \frac{d}{d x} [- 432 x^{5}] + \frac{d}{d x} [240 x^{4}]$

Step 2.3

Evaluate $\frac{d}{d x} [- 432 x^{5}]$ .

Tap for more steps...

Step 2.3.1

Since $- 432$ is constant with respect to $x$ , the derivative of $- 432 x^{5}$ with respect to $x$ is $- 432 \frac{d}{d x} [x^{5}]$ .

$1134 x^{5} - 432 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [240 x^{4}]$

Step 2.3.2

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

$1134 x^{5} - 432 (5 x^{4}) + \frac{d}{d x} [240 x^{4}]$

Step 2.3.3

Multiply $5$ by $- 432$ .

$1134 x^{5} - 2160 x^{4} + \frac{d}{d x} [240 x^{4}]$

Step 2.4

Evaluate $\frac{d}{d x} [240 x^{4}]$ .

Tap for more steps...

Step 2.4.1

Since $240$ is constant with respect to $x$ , the derivative of $240 x^{4}$ with respect to $x$ is $240 \frac{d}{d x} [x^{4}]$ .

$1134 x^{5} - 2160 x^{4} + 240 \frac{d}{d x} [x^{4}]$

Step 2.4.2

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

$1134 x^{5} - 2160 x^{4} + 240 (4 x^{3})$

Step 2.4.3

Multiply $4$ by $240$ .

$f'' (x) = 1134 x^{5} - 2160 x^{4} + 960 x^{3}$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $1134 x^{5} - 2160 x^{4} + 960 x^{3}$ with respect to $x$ is $\frac{d}{d x} [1134 x^{5}] + \frac{d}{d x} [- 2160 x^{4}] + \frac{d}{d x} [960 x^{3}]$ .

$\frac{d}{d x} [1134 x^{5}] + \frac{d}{d x} [- 2160 x^{4}] + \frac{d}{d x} [960 x^{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [1134 x^{5}]$ .

Tap for more steps...

Step 3.2.1

Since $1134$ is constant with respect to $x$ , the derivative of $1134 x^{5}$ with respect to $x$ is $1134 \frac{d}{d x} [x^{5}]$ .

$1134 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2160 x^{4}] + \frac{d}{d x} [960 x^{3}]$

Step 3.2.2

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

$1134 (5 x^{4}) + \frac{d}{d x} [- 2160 x^{4}] + \frac{d}{d x} [960 x^{3}]$

Step 3.2.3

Multiply $5$ by $1134$ .

$5670 x^{4} + \frac{d}{d x} [- 2160 x^{4}] + \frac{d}{d x} [960 x^{3}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 2160 x^{4}]$ .

Tap for more steps...

Step 3.3.1

Since $- 2160$ is constant with respect to $x$ , the derivative of $- 2160 x^{4}$ with respect to $x$ is $- 2160 \frac{d}{d x} [x^{4}]$ .

$5670 x^{4} - 2160 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [960 x^{3}]$

Step 3.3.2

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

$5670 x^{4} - 2160 (4 x^{3}) + \frac{d}{d x} [960 x^{3}]$

Step 3.3.3

Multiply $4$ by $- 2160$ .

$5670 x^{4} - 8640 x^{3} + \frac{d}{d x} [960 x^{3}]$

Step 3.4

Evaluate $\frac{d}{d x} [960 x^{3}]$ .

Tap for more steps...

Step 3.4.1

Since $960$ is constant with respect to $x$ , the derivative of $960 x^{3}$ with respect to $x$ is $960 \frac{d}{d x} [x^{3}]$ .

$5670 x^{4} - 8640 x^{3} + 960 \frac{d}{d x} [x^{3}]$

Step 3.4.2

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

$5670 x^{4} - 8640 x^{3} + 960 (3 x^{2})$

Step 3.4.3

Multiply $3$ by $960$ .

$f''' (x) = 5670 x^{4} - 8640 x^{3} + 2880 x^{2}$

Step 4

Find the fourth derivative.

Tap for more steps...

Step 4.1

By the Sum Rule, the derivative of $5670 x^{4} - 8640 x^{3} + 2880 x^{2}$ with respect to $x$ is $\frac{d}{d x} [5670 x^{4}] + \frac{d}{d x} [- 8640 x^{3}] + \frac{d}{d x} [2880 x^{2}]$ .

$\frac{d}{d x} [5670 x^{4}] + \frac{d}{d x} [- 8640 x^{3}] + \frac{d}{d x} [2880 x^{2}]$

Step 4.2

Evaluate $\frac{d}{d x} [5670 x^{4}]$ .

Tap for more steps...

Step 4.2.1

Since $5670$ is constant with respect to $x$ , the derivative of $5670 x^{4}$ with respect to $x$ is $5670 \frac{d}{d x} [x^{4}]$ .

$5670 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8640 x^{3}] + \frac{d}{d x} [2880 x^{2}]$

Step 4.2.2

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

$5670 (4 x^{3}) + \frac{d}{d x} [- 8640 x^{3}] + \frac{d}{d x} [2880 x^{2}]$

Step 4.2.3

Multiply $4$ by $5670$ .

$22680 x^{3} + \frac{d}{d x} [- 8640 x^{3}] + \frac{d}{d x} [2880 x^{2}]$

Step 4.3

Evaluate $\frac{d}{d x} [- 8640 x^{3}]$ .

Tap for more steps...

Step 4.3.1

Since $- 8640$ is constant with respect to $x$ , the derivative of $- 8640 x^{3}$ with respect to $x$ is $- 8640 \frac{d}{d x} [x^{3}]$ .

$22680 x^{3} - 8640 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [2880 x^{2}]$

Step 4.3.2

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

$22680 x^{3} - 8640 (3 x^{2}) + \frac{d}{d x} [2880 x^{2}]$

Step 4.3.3

Multiply $3$ by $- 8640$ .

$22680 x^{3} - 25920 x^{2} + \frac{d}{d x} [2880 x^{2}]$

Step 4.4

Evaluate $\frac{d}{d x} [2880 x^{2}]$ .

Tap for more steps...

Step 4.4.1

Since $2880$ is constant with respect to $x$ , the derivative of $2880 x^{2}$ with respect to $x$ is $2880 \frac{d}{d x} [x^{2}]$ .

$22680 x^{3} - 25920 x^{2} + 2880 \frac{d}{d x} [x^{2}]$

Step 4.4.2

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

$22680 x^{3} - 25920 x^{2} + 2880 (2 x)$

Step 4.4.3

Multiply $2$ by $2880$ .

$f^{4} (x) = 22680 x^{3} - 25920 x^{2} + 5760 x$