Calculus Examples

$h (u) = u^{3} {(9 u^{2} - 6)}^{4}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d u} [f (u) g (u)]$ is $f (u) \frac{d}{d u} [g (u)] + g (u) \frac{d}{d u} [f (u)]$ where $f (u) = u^{3}$ and $g (u) = {(9 u^{2} - 6)}^{4}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = u^{4}$ and $g (u) = 9 u^{2} - 6$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $9 u^{2} - 6$ .

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

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 $n = 4$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with $9 u^{2} - 6$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $9 u^{2} - 6$ with respect to $u$ is $\frac{d}{d u} [9 u^{2}] + \frac{d}{d u} [- 6]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $9$ .

$u^{3} (4 {(9 u^{2} - 6)}^{3} (18 u + \frac{d}{d u} [- 6])) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 3.5

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

$u^{3} (4 {(9 u^{2} - 6)}^{3} (18 u + 0)) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $18 u$ and $0$ .

$u^{3} (4 {(9 u^{2} - 6)}^{3} (18 u)) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 3.6.2

Multiply $18$ by $4$ .

$u^{3} (72 {(9 u^{2} - 6)}^{3} u) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 4

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

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Step 4.1

Move $u$ .

$u \cdot u^{3} (72 {(9 u^{2} - 6)}^{3}) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 4.2

Multiply $u$ by $u^{3}$ .

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Step 4.2.1

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

$u^{1} u^{3} (72 {(9 u^{2} - 6)}^{3}) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 4.2.2

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

$u^{1 + 3} (72 {(9 u^{2} - 6)}^{3}) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 4.3

Add $1$ and $3$ .

$u^{4} (72 {(9 u^{2} - 6)}^{3}) + {(9 u^{2} - 6)}^{4} \frac{d}{d u} [u^{3}]$

Step 5

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

$u^{4} (72 {(9 u^{2} - 6)}^{3}) + {(9 u^{2} - 6)}^{4} (3 u^{2})$

Step 6

Move $3$ to the left of ${(9 u^{2} - 6)}^{4}$ .

$u^{4} (72 {(9 u^{2} - 6)}^{3}) + 3 \cdot {(9 u^{2} - 6)}^{4} u^{2}$

Step 7

Simplify.

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Step 7.1

Factor $u^{2} \cdot 3 {(9 u^{2} - 6)}^{3}$ out of $u^{4} (72 {(9 u^{2} - 6)}^{3}) + 3 {(9 u^{2} - 6)}^{4} u^{2}$ .

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Step 7.1.1

Factor $u^{2} \cdot 3 {(9 u^{2} - 6)}^{3}$ out of $u^{4} (72 {(9 u^{2} - 6)}^{3})$ .

$u^{2} \cdot 3 {(9 u^{2} - 6)}^{3} (u^{2} \cdot (24)) + 3 {(9 u^{2} - 6)}^{4} u^{2}$

Step 7.1.2

Factor $u^{2} \cdot 3 {(9 u^{2} - 6)}^{3}$ out of $3 {(9 u^{2} - 6)}^{4} u^{2}$ .

$u^{2} \cdot 3 {(9 u^{2} - 6)}^{3} (u^{2} \cdot (24)) + u^{2} \cdot 3 {(9 u^{2} - 6)}^{3} (9 u^{2} - 6)$

Step 7.1.3

Factor $u^{2} \cdot 3 {(9 u^{2} - 6)}^{3}$ out of $u^{2} \cdot 3 {(9 u^{2} - 6)}^{3} (u^{2} \cdot (24)) + u^{2} \cdot 3 {(9 u^{2} - 6)}^{3} (9 u^{2} - 6)$ .

$u^{2} \cdot 3 {(9 u^{2} - 6)}^{3} (u^{2} \cdot (24) + 9 u^{2} - 6)$

Step 7.2

Combine terms.

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Step 7.2.1

Move $3$ to the left of $u^{2}$ .

$3 \cdot u^{2} {(9 u^{2} - 6)}^{3} (u^{2} \cdot (24) + 9 u^{2} - 6)$

Step 7.2.2

Move $24$ to the left of $u^{2}$ .

$3 u^{2} {(9 u^{2} - 6)}^{3} (24 \cdot u^{2} + 9 u^{2} - 6)$

Step 7.2.3

Add $24 u^{2}$ and $9 u^{2}$ .

$3 u^{2} {(9 u^{2} - 6)}^{3} (33 u^{2} - 6)$