Calculus Examples

$y = 8 x^{- 7} + 2 x^{5} - \frac{5}{x^{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (8 x^{- 7} + 2 x^{5} - \frac{5}{x^{3}})$

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 $8 x^{- 7} + 2 x^{5} - \frac{5}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [8 x^{- 7}] + \frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- \frac{5}{x^{3}}]$ .

$\frac{d}{d x} [8 x^{- 7}] + \frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.2

Evaluate $\frac{d}{d x} [8 x^{- 7}]$ .

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Step 3.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{- 7}$ with respect to $x$ is $8 \frac{d}{d x} [x^{- 7}]$ .

$8 \frac{d}{d x} [x^{- 7}] + \frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.2.2

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

$8 (- 7 x^{- 8}) + \frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.2.3

Multiply $- 7$ by $8$ .

$- 56 x^{- 8} + \frac{d}{d x} [2 x^{5}] + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.3

Evaluate $\frac{d}{d x} [2 x^{5}]$ .

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Step 3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{5}$ with respect to $x$ is $2 \frac{d}{d x} [x^{5}]$ .

$- 56 x^{- 8} + 2 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.3.2

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

$- 56 x^{- 8} + 2 (5 x^{4}) + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.3.3

Multiply $5$ by $2$ .

$- 56 x^{- 8} + 10 x^{4} + \frac{d}{d x} [- \frac{5}{x^{3}}]$

Step 3.4

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

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Step 3.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x^{3}}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$- 56 x^{- 8} + 10 x^{4} - 5 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.4.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$- 56 x^{- 8} + 10 x^{4} - 5 \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 3.4.3

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 $g (x) = x^{3}$ .

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Step 3.4.3.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$- 56 x^{- 8} + 10 x^{4} - 5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}])$

Step 3.4.3.2

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

$- 56 x^{- 8} + 10 x^{4} - 5 (- u^{- 2} \frac{d}{d x} [x^{3}])$

Step 3.4.3.3

Replace all occurrences of $u$ with $x^{3}$ .

$- 56 x^{- 8} + 10 x^{4} - 5 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 3.4.4

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

$- 56 x^{- 8} + 10 x^{4} - 5 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 3.4.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 3.4.5.1

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

$- 56 x^{- 8} + 10 x^{4} - 5 (- x^{3 \cdot - 2} (3 x^{2}))$

Step 3.4.5.2

Multiply $3$ by $- 2$ .

$- 56 x^{- 8} + 10 x^{4} - 5 (- x^{- 6} (3 x^{2}))$

Step 3.4.6

Multiply $3$ by $- 1$ .

$- 56 x^{- 8} + 10 x^{4} - 5 (- 3 x^{- 6} x^{2})$

Step 3.4.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.4.7.1

Move $x^{2}$ .

$- 56 x^{- 8} + 10 x^{4} - 5 (- 3 (x^{2} x^{- 6}))$

Step 3.4.7.2

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

$- 56 x^{- 8} + 10 x^{4} - 5 (- 3 x^{2 - 6})$

Step 3.4.7.3

Subtract $6$ from $2$ .

$- 56 x^{- 8} + 10 x^{4} - 5 (- 3 x^{- 4})$

Step 3.4.8

Multiply $- 3$ by $- 5$ .

$- 56 x^{- 8} + 10 x^{4} + 15 x^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$- 56 \frac{1}{x^{8}} + 10 x^{4} + 15 x^{- 4}$

Step 3.5.2

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

$- 56 \frac{1}{x^{8}} + 10 x^{4} + 15 \frac{1}{x^{4}}$

Step 3.5.3

Combine terms.

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Step 3.5.3.1

Combine $- 56$ and $\frac{1}{x^{8}}$ .

$\frac{- 56}{x^{8}} + 10 x^{4} + 15 \frac{1}{x^{4}}$

Step 3.5.3.2

Move the negative in front of the fraction.

$- \frac{56}{x^{8}} + 10 x^{4} + 15 \frac{1}{x^{4}}$

Step 3.5.3.3

Combine $15$ and $\frac{1}{x^{4}}$ .

$- \frac{56}{x^{8}} + 10 x^{4} + \frac{15}{x^{4}}$

Step 3.5.4

Reorder terms.

$10 x^{4} + \frac{15}{x^{4}} - \frac{56}{x^{8}}$

Step 4

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

$y' = 10 x^{4} + \frac{15}{x^{4}} - \frac{56}{x^{8}}$

Step 5

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

$\frac{d y}{d x} = 10 x^{4} + \frac{15}{x^{4}} - \frac{56}{x^{8}}$