Calculus Examples

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

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Raise $4$ to the power of $3$ .

$\frac{d}{d x} [2 x^{4} - 1 \cdot 64 + 6 x]$

Step 1.2

By the Sum Rule, the derivative of $2 x^{4} - 1 \cdot 64 + 6 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [- 1 \cdot 64] + \frac{d}{d x} [6 x]$ .

$\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [- 1 \cdot 64] + \frac{d}{d x} [6 x]$

Step 2

Evaluate $\frac{d}{d x} [2 x^{4}]$ .

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Step 2.1

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

$2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 1 \cdot 64] + \frac{d}{d x} [6 x]$

Step 2.2

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

$2 (4 x^{3}) + \frac{d}{d x} [- 1 \cdot 64] + \frac{d}{d x} [6 x]$

Step 2.3

Multiply $4$ by $2$ .

$8 x^{3} + \frac{d}{d x} [- 1 \cdot 64] + \frac{d}{d x} [6 x]$

Step 3

Evaluate $\frac{d}{d x} [- 1 \cdot 64]$ .

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Step 3.1

Multiply $- 1$ by $64$ .

$8 x^{3} + \frac{d}{d x} [- 64] + \frac{d}{d x} [6 x]$

Step 3.2

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64$ with respect to $x$ is $0$ .

$8 x^{3} + 0 + \frac{d}{d x} [6 x]$

Step 4

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

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Step 4.1

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

$8 x^{3} + 0 + 6 \frac{d}{d x} [x]$

Step 4.2

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

$8 x^{3} + 0 + 6 \cdot 1$

Step 4.3

Multiply $6$ by $1$ .

$8 x^{3} + 0 + 6$

Step 5

Add $8 x^{3}$ and $0$ .

$8 x^{3} + 6$