Calculus Examples

$y = 3 x^{5} - 2 x^{- 3} + \sqrt[4]{x^{7}}$

Step 1

By the Sum Rule, the derivative of $3 x^{5} - 2 x^{- 3} + \sqrt[4]{x^{7}}$ with respect to $x$ is $\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 2 x^{- 3}] + \frac{d}{d x} [\sqrt[4]{x^{7}}]$ .

$\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [- 2 x^{- 3}] + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 2

Evaluate $\frac{d}{d x} [3 x^{5}]$ .

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2 x^{- 3}] + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$3 (5 x^{4}) + \frac{d}{d x} [- 2 x^{- 3}] + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 2.3

Multiply $5$ by $3$ .

$15 x^{4} + \frac{d}{d x} [- 2 x^{- 3}] + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 3

Evaluate $\frac{d}{d x} [- 2 x^{- 3}]$ .

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{- 3}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{- 3}]$ .

$15 x^{4} - 2 \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$15 x^{4} - 2 (- 3 x^{- 4}) + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 3.3

Multiply $- 3$ by $- 2$ .

$15 x^{4} + 6 x^{- 4} + \frac{d}{d x} [\sqrt[4]{x^{7}}]$

Step 4

Evaluate $\frac{d}{d x} [\sqrt[4]{x^{7}}]$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{7}}$ as $x^{\frac{7}{4}}$ .

$15 x^{4} + 6 x^{- 4} + \frac{d}{d x} [x^{\frac{7}{4}}]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{4}$ .

$15 x^{4} + 6 x^{- 4} + \frac{7}{4} x^{\frac{7}{4} - 1}$

Step 4.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$15 x^{4} + 6 x^{- 4} + \frac{7}{4} x^{\frac{7}{4} - 1 \cdot \frac{4}{4}}$

Step 4.4

Combine $- 1$ and $\frac{4}{4}$ .

$15 x^{4} + 6 x^{- 4} + \frac{7}{4} x^{\frac{7}{4} + \frac{- 1 \cdot 4}{4}}$

Step 4.5

Combine the numerators over the common denominator.

$15 x^{4} + 6 x^{- 4} + \frac{7}{4} x^{\frac{7 - 1 \cdot 4}{4}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $4$ .

$15 x^{4} + 6 x^{- 4} + \frac{7}{4} x^{\frac{7 - 4}{4}}$

Step 4.6.2

Subtract $4$ from $7$ .

$15 x^{4} + 6 x^{- 4} + \frac{7}{4} x^{\frac{3}{4}}$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$15 x^{4} + 6 \frac{1}{x^{4}} + \frac{7}{4} x^{\frac{3}{4}}$

Step 5.2

Combine $6$ and $\frac{1}{x^{4}}$ .

$15 x^{4} + \frac{6}{x^{4}} + \frac{7}{4} x^{\frac{3}{4}}$

Step 5.3

Reorder terms.

$15 x^{4} + \frac{7}{4} x^{\frac{3}{4}} + \frac{6}{x^{4}}$

Step 5.4

Combine $\frac{7}{4}$ and $x^{\frac{3}{4}}$ .

$15 x^{4} + \frac{7 x^{\frac{3}{4}}}{4} + \frac{6}{x^{4}}$