Calculus Examples

$1 - {(\frac{10}{x})}^{4}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $1 - {(\frac{10}{x})}^{4}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- {(\frac{10}{x})}^{4}]$ .

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

Step 1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- {(\frac{10}{x})}^{4}]$

Step 2

Evaluate $\frac{d}{d x} [- {(\frac{10}{x})}^{4}]$ .

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(\frac{10}{x})}^{4}$ with respect to $x$ is $- \frac{d}{d x} [{(\frac{10}{x})}^{4}]$ .

$0 - \frac{d}{d x} [{(\frac{10}{x})}^{4}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = \frac{10}{x}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{10}{x}$ .

$0 - (\frac{d}{d u} [u^{4}] \frac{d}{d x} [\frac{10}{x}])$

Step 2.2.2

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

$0 - (4 u^{3} \frac{d}{d x} [\frac{10}{x}])$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{10}{x}$ .

$0 - (4 {(\frac{10}{x})}^{3} \frac{d}{d x} [\frac{10}{x}])$

Step 2.3

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10}{x}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{1}{x}]$ .

$0 - (4 {(\frac{10}{x})}^{3} (10 \frac{d}{d x} [\frac{1}{x}]))$

Step 2.4

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

$0 - (4 {(\frac{10}{x})}^{3} (10 \frac{d}{d x} [x^{- 1}]))$

Step 2.5

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

$0 - (4 {(\frac{10}{x})}^{3} (10 (- x^{- 2})))$

Step 2.6

Multiply $- 1$ by $10$ .

$0 - (4 {(\frac{10}{x})}^{3} (- 10 x^{- 2}))$

Step 2.7

Multiply $- 10$ by $4$ .

$0 - (- 40 {(\frac{10}{x})}^{3} x^{- 2})$

Step 2.8

Multiply $- 40$ by $- 1$ .

$0 + 40 {(\frac{10}{x})}^{3} x^{- 2}$

Step 3

Simplify.

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Step 3.1

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

$0 + 40 {(\frac{10}{x})}^{3} \frac{1}{x^{2}}$

Step 3.2

Apply the product rule to $\frac{10}{x}$ .

$0 + 40 \frac{10^{3}}{x^{3}} \frac{1}{x^{2}}$

Step 3.3

Combine terms.

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Step 3.3.1

Raise $10$ to the power of $3$ .

$0 + 40 \frac{1000}{x^{3}} \frac{1}{x^{2}}$

Step 3.3.2

Combine $40$ and $\frac{1000}{x^{3}}$ .

$0 + \frac{40 \cdot 1000}{x^{3}} \cdot \frac{1}{x^{2}}$

Step 3.3.3

Multiply $40$ by $1000$ .

$0 + \frac{40000}{x^{3}} \cdot \frac{1}{x^{2}}$

Step 3.3.4

Multiply $\frac{40000}{x^{3}}$ by $\frac{1}{x^{2}}$ .

$0 + \frac{40000}{x^{3} x^{2}}$

Step 3.3.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.3.5.1

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

$0 + \frac{40000}{x^{3 + 2}}$

Step 3.3.5.2

Add $3$ and $2$ .

$0 + \frac{40000}{x^{5}}$

Step 3.3.6

Add $0$ and $\frac{40000}{x^{5}}$ .

$\frac{40000}{x^{5}}$