Calculus Examples

$20 - \frac{20000000}{x^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $20 - \frac{20000000}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [20] + \frac{d}{d x} [- \frac{20000000}{x^{2}}]$ .

$\frac{d}{d x} [20] + \frac{d}{d x} [- \frac{20000000}{x^{2}}]$

Step 1.1.1.2

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{20000000}{x^{2}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- \frac{20000000}{x^{2}}]$ .

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Step 1.1.2.1

Since $- 20000000$ is constant with respect to $x$ , the derivative of $- \frac{20000000}{x^{2}}$ with respect to $x$ is $- 20000000 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$0 - 20000000 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.1.2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$0 - 20000000 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.1.2.3

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 $g (x) = x^{2}$ .

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Step 1.1.2.3.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$0 - 20000000 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2}])$

Step 1.1.2.3.2

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

$0 - 20000000 (- u^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.2.3.3

Replace all occurrences of $u$ with $x^{2}$ .

$0 - 20000000 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}])$

Step 1.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - 20000000 (- {(x^{2})}^{- 2} (2 x))$

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

$0 - 20000000 (- x^{2 \cdot - 2} (2 x))$

Step 1.1.2.5.2

Multiply $2$ by $- 2$ .

$0 - 20000000 (- x^{- 4} (2 x))$

Step 1.1.2.6

Multiply $2$ by $- 1$ .

$0 - 20000000 (- 2 x^{- 4} x)$

Step 1.1.2.7

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

$0 - 20000000 (- 2 (x^{1} x^{- 4}))$

Step 1.1.2.8

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

$0 - 20000000 (- 2 x^{1 - 4})$

Step 1.1.2.9

Subtract $4$ from $1$ .

$0 - 20000000 (- 2 x^{- 3})$

Step 1.1.2.10

Multiply $- 2$ by $- 20000000$ .

$0 + 40000000 x^{- 3}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

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

$0 + 40000000 \frac{1}{x^{3}}$

Step 1.1.3.2

Combine terms.

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Step 1.1.3.2.1

Combine $40000000$ and $\frac{1}{x^{3}}$ .

$0 + \frac{40000000}{x^{3}}$

Step 1.1.3.2.2

Add $0$ and $\frac{40000000}{x^{3}}$ .

$f' (x) = \frac{40000000}{x^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{40000000}{x^{3}}$ .

$\frac{40000000}{x^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{40000000}{x^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{40000000}{x^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$40000000 = 0$

Step 2.3

Since $40000000 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{40000000}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 4

Evaluate $20 - \frac{20000000}{x^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$20 - \frac{20000000}{{(0)}^{2}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Raising $0$ to any positive power yields $0$ .

$20 - \frac{20000000}{0}$

Step 4.1.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found