Calculus Examples

$f (x) = 2 x (4 x + 4^{5})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Raise $4$ to the power of $5$ .

$\frac{d}{d x} [2 x (4 x + 1024)]$

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x (4 x + 1024)$ with respect to $x$ is $2 \frac{d}{d x} [x (4 x + 1024)]$ .

$2 \frac{d}{d x} [x (4 x + 1024)]$

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 $f (x) = x$ and $g (x) = 4 x + 1024$ .

$2 (x \frac{d}{d x} [4 x + 1024] + (4 x + 1024) \frac{d}{d x} [x])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 x + 1024$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [1024]$ .

$2 (x (\frac{d}{d x} [4 x] + \frac{d}{d x} [1024]) + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.2

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

$2 (x (4 \frac{d}{d x} [x] + \frac{d}{d x} [1024]) + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.3

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

$2 (x (4 \cdot 1 + \frac{d}{d x} [1024]) + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.4

Multiply $4$ by $1$ .

$2 (x (4 + \frac{d}{d x} [1024]) + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.5

Since $1024$ is constant with respect to $x$ , the derivative of $1024$ with respect to $x$ is $0$ .

$2 (x (4 + 0) + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $4$ and $0$ .

$2 (x \cdot 4 + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.6.2

Move $4$ to the left of $x$ .

$2 (4 \cdot x + (4 x + 1024) \frac{d}{d x} [x])$

Step 3.7

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

$2 (4 x + (4 x + 1024) \cdot 1)$

Step 3.8

Simplify by adding terms.

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Step 3.8.1

Multiply $4 x + 1024$ by $1$ .

$2 (4 x + 4 x + 1024)$

Step 3.8.2

Add $4 x$ and $4 x$ .

$2 (8 x + 1024)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$2 (8 x) + 2 \cdot 1024$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $8$ by $2$ .

$16 x + 2 \cdot 1024$

Step 4.2.2

Multiply $2$ by $1024$ .

$16 x + 2048$