Calculus Examples

$g (x) = 3 {(x^{3} - 6)}^{2} {(x^{2} + 4 x - 6)}^{10}$

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 $f (x) = 3 {(x^{3} - 6)}^{2}$ and $g (x) = {(x^{2} + 4 x - 6)}^{10}$ .

$3 {(x^{3} - 6)}^{2} \frac{d}{d x} [{(x^{2} + 4 x - 6)}^{10}] + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{10}$ and $g (x) = x^{2} + 4 x - 6$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + 4 x - 6$ .

$3 {(x^{3} - 6)}^{2} (\frac{d}{d u_{1}} [{u_{1}}^{10}] \frac{d}{d x} [x^{2} + 4 x - 6]) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 10$ .

$3 {(x^{3} - 6)}^{2} (10 {u_{1}}^{9} \frac{d}{d x} [x^{2} + 4 x - 6]) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $x^{2} + 4 x - 6$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} \frac{d}{d x} [x^{2} + 4 x - 6]) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x^{2} + 4 x - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 6]$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 6])) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.2

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

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 6])) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.3

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

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.4

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

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4 \cdot 1 + \frac{d}{d x} [- 6])) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.5

Multiply $4$ by $1$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4 + \frac{d}{d x} [- 6])) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.6

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4 + 0)) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.7

Add $2 x + 4$ and $0$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} \frac{d}{d x} [3 {(x^{3} - 6)}^{2}]$

Step 3.8

Since $3$ is constant with respect to $x$ , the derivative of $3 {(x^{3} - 6)}^{2}$ with respect to $x$ is $3 \frac{d}{d x} [{(x^{3} - 6)}^{2}]$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (3 \frac{d}{d x} [{(x^{3} - 6)}^{2}])$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{3} - 6$ .

Tap for more steps...

Step 4.1

To apply the Chain Rule, set $u_{2}$ as $x^{3} - 6$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (3 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [x^{3} - 6]))$

Step 4.2

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

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (3 (2 u_{2} \frac{d}{d x} [x^{3} - 6]))$

Step 4.3

Replace all occurrences of $u_{2}$ with $x^{3} - 6$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (3 (2 (x^{3} - 6) \frac{d}{d x} [x^{3} - 6]))$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Multiply $2$ by $3$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (6 ((x^{3} - 6) \frac{d}{d x} [x^{3} - 6]))$

Step 5.2

By the Sum Rule, the derivative of $x^{3} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6]$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (6 (x^{3} - 6) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6]))$

Step 5.3

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

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (6 (x^{3} - 6) (3 x^{2} + \frac{d}{d x} [- 6]))$

Step 5.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (6 (x^{3} - 6) (3 x^{2} + 0))$

Step 5.5

Simplify the expression.

Tap for more steps...

Step 5.5.1

Add $3 x^{2}$ and $0$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (6 (x^{3} - 6) (3 x^{2}))$

Step 5.5.2

Multiply $3$ by $6$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 (x^{3} - 6) x^{2})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} ((18 x^{3} + 18 \cdot - 6) x^{2})$

Step 6.2

Apply the distributive property.

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{3} x^{2} + 18 \cdot - 6 x^{2})$

Step 6.3

Combine terms.

Tap for more steps...

Step 6.3.1

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

Tap for more steps...

Step 6.3.1.1

Move $x^{2}$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 (x^{2} x^{3}) + 18 \cdot - 6 x^{2})$

Step 6.3.1.2

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

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{2 + 3} + 18 \cdot - 6 x^{2})$

Step 6.3.1.3

Add $2$ and $3$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{5} + 18 \cdot - 6 x^{2})$

Step 6.3.2

Multiply $18$ by $- 6$ .

$3 {(x^{3} - 6)}^{2} (10 {(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{5} - 108 x^{2})$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Multiply $10$ by $3$ .

$30 {(x^{3} - 6)}^{2} ({(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{5} - 108 x^{2})$

Step 7.2

Factor ${(x^{2} + 4 x - 6)}^{9}$ out of $30 {(x^{3} - 6)}^{2} ({(x^{2} + 4 x - 6)}^{9} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{5} - 108 x^{2})$ .

Tap for more steps...

Step 7.2.1

Factor ${(x^{2} + 4 x - 6)}^{9}$ out of $30 {(x^{3} - 6)}^{2} ({(x^{2} + 4 x - 6)}^{9} (2 x + 4))$ .

${(x^{2} + 4 x - 6)}^{9} (30 {(x^{3} - 6)}^{2} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{10} (18 x^{5} - 108 x^{2})$

Step 7.2.2

Factor ${(x^{2} + 4 x - 6)}^{9}$ out of ${(x^{2} + 4 x - 6)}^{10} (18 x^{5} - 108 x^{2})$ .

${(x^{2} + 4 x - 6)}^{9} (30 {(x^{3} - 6)}^{2} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{9} ((x^{2} + 4 x - 6) (18 x^{5} - 108 x^{2}))$

Step 7.2.3

Factor ${(x^{2} + 4 x - 6)}^{9}$ out of ${(x^{2} + 4 x - 6)}^{9} (30 {(x^{3} - 6)}^{2} (2 x + 4)) + {(x^{2} + 4 x - 6)}^{9} ((x^{2} + 4 x - 6) (18 x^{5} - 108 x^{2}))$ .

${(x^{2} + 4 x - 6)}^{9} (30 {(x^{3} - 6)}^{2} (2 x + 4) + (x^{2} + 4 x - 6) (18 x^{5} - 108 x^{2}))$