Calculus Examples

$\frac{1}{x^{9}}$

Find the first derivative.

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Apply basic rules of exponents.

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Rewrite $\frac{1}{x^{9}}$ as ${(x^{9})}^{- 1}$ .

$f' (x) = \frac{d}{d x} ({(x^{9})}^{- 1})$

Multiply the exponents in ${(x^{9})}^{- 1}$ .

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

$f' (x) = \frac{d}{d x} (x^{9 \cdot - 1})$

Multiply $9$ by $- 1$ .

$f' (x) = \frac{d}{d x} (x^{- 9})$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 9$ .

$f' (x) = - 9 x^{- 10}$

Simplify.

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

$f' (x) = - 9 \frac{1}{x^{10}}$

Combine terms.

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Combine $- 9$ and $\frac{1}{x^{10}}$ .

$f' (x) = \frac{- 9}{x^{10}}$

Move the negative in front of the fraction.

$f' (x) = - \frac{9}{x^{10}}$

Find the second derivative.

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Since $- 9$ is constant with respect to $x$ , the derivative of $- \frac{9}{x^{10}}$ with respect to $x$ is $- 9 \frac{d}{d x} [\frac{1}{x^{10}}]$ .

$f'' (x) = - 9 \frac{d}{d x} \frac{1}{x^{10}}$

Apply basic rules of exponents.

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Rewrite $\frac{1}{x^{10}}$ as ${(x^{10})}^{- 1}$ .

$f'' (x) = - 9 \frac{d}{d x} {(x^{10})}^{- 1}$

Multiply the exponents in ${(x^{10})}^{- 1}$ .

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

$f'' (x) = - 9 \frac{d}{d x} x^{10 \cdot - 1}$

Multiply $10$ by $- 1$ .

$f'' (x) = - 9 \frac{d}{d x} x^{- 10}$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 10$ .

$f'' (x) = - 9 (- 10 x^{- 11})$

Multiply $- 10$ by $- 9$ .

$f'' (x) = 90 x^{- 11}$

Simplify.

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

$f'' (x) = 90 (\frac{1}{x^{11}})$

Combine $90$ and $\frac{1}{x^{11}}$ .

$f'' (x) = \frac{90}{x^{11}}$