Calculus Examples

$y = 3 e^{x} + \frac{4}{\sqrt[3]{x}}$

Step 1

Multiply $\frac{4}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$\frac{d}{d y} [3 e^{x} + \frac{4}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}]$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{4}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}]$

Step 2.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}]$

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}]$

Step 2.4

Add $1$ and $2$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}]$

Step 2.5

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}]$

Step 2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}]$

Step 2.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}]$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}]$

Step 2.5.4.2

Rewrite the expression.

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{x^{1}}]$

Step 2.5.5

Simplify.

$\frac{d}{d y} [3 e^{x} + \frac{4 {\sqrt[3]{x}}^{2}}{x}]$

Step 3

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$\frac{d}{d y} [3 e^{x} + \frac{4 \sqrt[3]{x^{2}}}{x}]$

Step 4

By the Sum Rule, the derivative of $3 e^{x} + \frac{4 \sqrt[3]{x^{2}}}{x}$ with respect to $y$ is $\frac{d}{d y} [3 e^{x}] + \frac{d}{d y} [\frac{4 \sqrt[3]{x^{2}}}{x}]$ .

$\frac{d}{d y} [3 e^{x}] + \frac{d}{d y} [\frac{4 \sqrt[3]{x^{2}}}{x}]$

Step 5

Since $3 e^{x}$ is constant with respect to $y$ , the derivative of $3 e^{x}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [\frac{4 \sqrt[3]{x^{2}}}{x}]$

Step 6

Since $\frac{4 \sqrt[3]{x^{2}}}{x}$ is constant with respect to $y$ , the derivative of $\frac{4 \sqrt[3]{x^{2}}}{x}$ with respect to $y$ is $0$ .

$0 + 0$

Step 7

Add $0$ and $0$ .

$0$