Calculus Examples

$h (x) = 7 x^{2} (6 x + 9^{4})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Raise $9$ to the power of $4$ .

$\frac{d}{d x} [7 x^{2} (6 x + 6561)]$

Step 1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2} (6 x + 6561)$ with respect to $x$ is $7 \frac{d}{d x} [x^{2} (6 x + 6561)]$ .

$7 \frac{d}{d x} [x^{2} (6 x + 6561)]$

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 $g (x) = 6 x + 6561$ .

$7 (x^{2} \frac{d}{d x} [6 x + 6561] + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $6 x + 6561$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [6561]$ .

$7 (x^{2} (\frac{d}{d x} [6 x] + \frac{d}{d x} [6561]) + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.2

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

$7 (x^{2} (6 \frac{d}{d x} [x] + \frac{d}{d x} [6561]) + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.3

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

$7 (x^{2} (6 \cdot 1 + \frac{d}{d x} [6561]) + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.4

Multiply $6$ by $1$ .

$7 (x^{2} (6 + \frac{d}{d x} [6561]) + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.5

Since $6561$ is constant with respect to $x$ , the derivative of $6561$ with respect to $x$ is $0$ .

$7 (x^{2} (6 + 0) + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $6$ and $0$ .

$7 (x^{2} \cdot 6 + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.6.2

Move $6$ to the left of $x^{2}$ .

$7 (6 \cdot x^{2} + (6 x + 6561) \frac{d}{d x} [x^{2}])$

Step 3.7

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

$7 (6 x^{2} + (6 x + 6561) (2 x))$

Step 3.8

Move $2$ to the left of $6 x + 6561$ .

$7 (6 x^{2} + 2 \cdot (6 x + 6561) x)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$7 (6 x^{2} + (2 (6 x) + 2 \cdot 6561) x)$

Step 4.2

Apply the distributive property.

$7 (6 x^{2} + 2 (6 x) x + 2 \cdot 6561 x)$

Step 4.3

Apply the distributive property.

$7 (6 x^{2}) + 7 (2 (6 x) x) + 7 (2 \cdot 6561 x)$

Step 4.4

Combine terms.

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Step 4.4.1

Multiply $6$ by $7$ .

$42 x^{2} + 7 (2 (6 x) x) + 7 (2 \cdot 6561 x)$

Step 4.4.2

Multiply $6$ by $2$ .

$42 x^{2} + 7 (12 x \cdot x) + 7 (2 \cdot 6561 x)$

Step 4.4.3

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

$42 x^{2} + 7 (12 (x^{1} x)) + 7 (2 \cdot 6561 x)$

Step 4.4.4

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

$42 x^{2} + 7 (12 (x^{1} x^{1})) + 7 (2 \cdot 6561 x)$

Step 4.4.5

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

$42 x^{2} + 7 (12 x^{1 + 1}) + 7 (2 \cdot 6561 x)$

Step 4.4.6

Add $1$ and $1$ .

$42 x^{2} + 7 (12 x^{2}) + 7 (2 \cdot 6561 x)$

Step 4.4.7

Multiply $12$ by $7$ .

$42 x^{2} + 84 x^{2} + 7 (2 \cdot 6561 x)$

Step 4.4.8

Multiply $2$ by $6561$ .

$42 x^{2} + 84 x^{2} + 7 (13122 x)$

Step 4.4.9

Multiply $13122$ by $7$ .

$42 x^{2} + 84 x^{2} + 91854 x$

Step 4.4.10

Add $42 x^{2}$ and $84 x^{2}$ .

$126 x^{2} + 91854 x$