Calculus Examples

$\frac{d}{d x} \cdot y = 4 x^{7} + x^{- \frac{1}{5}}$

Step 1

Factor each term.

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Step 1.1

Multiply $\frac{d}{d x}$ by $y$ .

$\frac{d}{d x} y = 4 x^{7} + x^{- \frac{1}{5}}$

Step 1.2

Combine $\frac{d}{d x}$ and $y$ .

$\frac{d y}{d x} = 4 x^{7} + x^{- \frac{1}{5}}$

Step 1.3

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

$\frac{d y}{d x} = 4 x^{7} + \frac{1}{x^{\frac{1}{5}}}$

Step 1.4

Reduce the expression $\frac{d y}{d x}$ by cancelling the common factors.

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Step 1.4.1

Cancel the common factor.

$\frac{d y}{d x} = 4 x^{7} + \frac{1}{x^{\frac{1}{5}}}$

Step 1.4.2

Rewrite the expression.

$\frac{y}{x} = 4 x^{7} + \frac{1}{x^{\frac{1}{5}}}$

Step 2

Solve the equation.

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Step 2.1

Multiply both sides by $x$ .

$\frac{y}{x} x = (4 x^{7} + \frac{1}{x^{\frac{1}{5}}}) x$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $x$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{y}{x} x = (4 x^{7} + \frac{1}{x^{\frac{1}{5}}}) x$

Step 2.2.1.1.2

Rewrite the expression.

$y = (4 x^{7} + \frac{1}{x^{\frac{1}{5}}}) x$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $(4 x^{7} + \frac{1}{x^{\frac{1}{5}}}) x$ .

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Step 2.2.2.1.1

Apply the distributive property.

$y = 4 x^{7} x + \frac{1}{x^{\frac{1}{5}}} x$

Step 2.2.2.1.2

Multiply $x^{7}$ by $x$ by adding the exponents.

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Step 2.2.2.1.2.1

Move $x$ .

$y = 4 (x \cdot x^{7}) + \frac{1}{x^{\frac{1}{5}}} x$

Step 2.2.2.1.2.2

Multiply $x$ by $x^{7}$ .

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Step 2.2.2.1.2.2.1

Raise $x$ to the power of $1$ .

$y = 4 (x^{1} x^{7}) + \frac{1}{x^{\frac{1}{5}}} x$

Step 2.2.2.1.2.2.2

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

$y = 4 x^{1 + 7} + \frac{1}{x^{\frac{1}{5}}} x$

Step 2.2.2.1.2.3

Add $1$ and $7$ .

$y = 4 x^{8} + \frac{1}{x^{\frac{1}{5}}} x$

Step 2.2.2.1.3

Combine $\frac{1}{x^{\frac{1}{5}}}$ and $x$ .

$y = 4 x^{8} + \frac{x}{x^{\frac{1}{5}}}$

Step 2.2.2.1.4

Simplify each term.

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Step 2.2.2.1.4.1

Move $x^{\frac{1}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$y = 4 x^{8} + x \cdot x^{- \frac{1}{5}}$

Step 2.2.2.1.4.2

Multiply $x$ by $x^{- \frac{1}{5}}$ by adding the exponents.

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Step 2.2.2.1.4.2.1

Multiply $x$ by $x^{- \frac{1}{5}}$ .

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Step 2.2.2.1.4.2.1.1

Raise $x$ to the power of $1$ .

$y = 4 x^{8} + x^{1} x^{- \frac{1}{5}}$

Step 2.2.2.1.4.2.1.2

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

$y = 4 x^{8} + x^{1 - \frac{1}{5}}$

Step 2.2.2.1.4.2.2

Write $1$ as a fraction with a common denominator.

$y = 4 x^{8} + x^{\frac{5}{5} - \frac{1}{5}}$

Step 2.2.2.1.4.2.3

Combine the numerators over the common denominator.

$y = 4 x^{8} + x^{\frac{5 - 1}{5}}$

Step 2.2.2.1.4.2.4

Subtract $1$ from $5$ .

$y = 4 x^{8} + x^{\frac{4}{5}}$

Step 3

The variable $d$ got canceled.

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$