Calculus Examples

$y = \frac{7}{x^{5}} - \frac{2}{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{7}{x^{5}} - \frac{2}{x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Evaluate $\frac{d}{d x} [\frac{7}{x^{5}}]$ .

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Step 3.2.1

Since $7$ is constant with respect to $x$ , the derivative of $\frac{7}{x^{5}}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$7 \frac{d}{d x} [\frac{1}{x^{5}}] + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.2

Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

$7 \frac{d}{d x} [{(x^{5})}^{- 1}] + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{5}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u$ as $x^{5}$ .

$7 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.3.2

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

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

Step 3.2.3.3

Replace all occurrences of $u$ with $x^{5}$ .

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

Step 3.2.4

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

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

Step 3.2.5

Multiply the exponents in ${(x^{5})}^{- 2}$ .

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Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $5$ by $- 2$ .

$7 (- x^{- 10} (5 x^{4})) + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.6

Multiply $5$ by $- 1$ .

$7 (- 5 x^{- 10} x^{4}) + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.7

Multiply $x^{- 10}$ by $x^{4}$ by adding the exponents.

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Step 3.2.7.1

Move $x^{4}$ .

$7 (- 5 (x^{4} x^{- 10})) + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.7.2

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

$7 (- 5 x^{4 - 10}) + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.7.3

Subtract $10$ from $4$ .

$7 (- 5 x^{- 6}) + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.2.8

Multiply $- 5$ by $7$ .

$- 35 x^{- 6} + \frac{d}{d x} [- \frac{2}{x}]$

Step 3.3

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

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Step 3.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x}]$ .

$- 35 x^{- 6} - 2 \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$- 35 x^{- 6} - 2 \frac{d}{d x} [x^{- 1}]$

Step 3.3.3

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

$- 35 x^{- 6} - 2 (- x^{- 2})$

Step 3.3.4

Multiply $- 1$ by $- 2$ .

$- 35 x^{- 6} + 2 x^{- 2}$

Step 3.4

Simplify.

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Step 3.4.1

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

$- 35 \frac{1}{x^{6}} + 2 x^{- 2}$

Step 3.4.2

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

$- 35 \frac{1}{x^{6}} + 2 \frac{1}{x^{2}}$

Step 3.4.3

Combine terms.

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Step 3.4.3.1

Combine $- 35$ and $\frac{1}{x^{6}}$ .

$\frac{- 35}{x^{6}} + 2 \frac{1}{x^{2}}$

Step 3.4.3.2

Move the negative in front of the fraction.

$- \frac{35}{x^{6}} + 2 \frac{1}{x^{2}}$

Step 3.4.3.3

Combine $2$ and $\frac{1}{x^{2}}$ .

$- \frac{35}{x^{6}} + \frac{2}{x^{2}}$

Step 3.4.4

Reorder terms.

$\frac{2}{x^{2}} - \frac{35}{x^{6}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{2}{x^{2}} - \frac{35}{x^{6}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{2}{x^{2}} - \frac{35}{x^{6}}$