Calculus Examples

$\sqrt[3]{x^{9.6}} + 2 e^{1.3}$

Step 1

By the Sum Rule, the derivative of $\sqrt[3]{x^{9.6}} + 2 e^{1.3}$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{x^{9.6}}] + \frac{d}{d x} [2 e^{1.3}]$ .

$\frac{d}{d x} [\sqrt[3]{x^{9.6}}] + \frac{d}{d x} [2 e^{1.3}]$

Step 2

Evaluate $\frac{d}{d x} [\sqrt[3]{x^{9.6}}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{9.6}}$ as $x^{\frac{9.6}{3}}$ .

$\frac{d}{d x} [x^{\frac{9.6}{3}}] + \frac{d}{d x} [2 e^{1.3}]$

Step 2.2

Divide $9.6$ by $3$ .

$\frac{d}{d x} [x^{3.2}] + \frac{d}{d x} [2 e^{1.3}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3.2$ .

$3.2 x^{2.2} + \frac{d}{d x} [2 e^{1.3}]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $2 e^{1.3}$ is constant with respect to $x$ , the derivative of $2 e^{1.3}$ with respect to $x$ is $0$ .

$3.2 x^{2.2} + 0$

Step 3.2

Add $3.2 x^{2.2}$ and $0$ .

$3.2 x^{2.2}$