Calculus Examples

$G (x) \sqrt[5]{x^{7} + 2 x}$

Step 1

Factor $x$ out of $x^{7} + 2 x$ .

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Step 1.1

Factor $x$ out of $x^{7}$ .

$\frac{d}{d G} [G (x) \sqrt[5]{x \cdot x^{6} + 2 x}]$

Step 1.2

Factor $x$ out of $2 x$ .

$\frac{d}{d G} [G (x) \sqrt[5]{x \cdot x^{6} + x \cdot 2}]$

Step 1.3

Factor $x$ out of $x \cdot x^{6} + x \cdot 2$ .

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

Step 2

Since $x \sqrt[5]{x (x^{6} + 2)}$ is constant with respect to $G$ , the derivative of $G x \sqrt[5]{x (x^{6} + 2)}$ with respect to $G$ is $x \sqrt[5]{x (x^{6} + 2)} \frac{d}{d G} [G]$ .

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

Step 3

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

$x \sqrt[5]{x (x^{6} + 2)} \cdot 1$

Step 4

Multiply $x$ by $1$ .

$x \sqrt[5]{x (x^{6} + 2)}$