Calculus Examples

$y = 8 x^{6} + 2 x^{17}$

Step 1

By the Sum Rule, the derivative of $8 x^{6} + 2 x^{17}$ with respect to $x$ is $\frac{d}{d x} [8 x^{6}] + \frac{d}{d x} [2 x^{17}]$ .

$\frac{d}{d x} [8 x^{6}] + \frac{d}{d x} [2 x^{17}]$

Step 2

Evaluate $\frac{d}{d x} [8 x^{6}]$ .

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Step 2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{6}$ with respect to $x$ is $8 \frac{d}{d x} [x^{6}]$ .

$8 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [2 x^{17}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$8 (6 x^{5}) + \frac{d}{d x} [2 x^{17}]$

Step 2.3

Multiply $6$ by $8$ .

$48 x^{5} + \frac{d}{d x} [2 x^{17}]$

Step 3

Evaluate $\frac{d}{d x} [2 x^{17}]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{17}$ with respect to $x$ is $2 \frac{d}{d x} [x^{17}]$ .

$48 x^{5} + 2 \frac{d}{d x} [x^{17}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 17$ .

$48 x^{5} + 2 (17 x^{16})$

Step 3.3

Multiply $17$ by $2$ .

$48 x^{5} + 34 x^{16}$

Step 4

Reorder terms.

$34 x^{16} + 48 x^{5}$