Calculus Examples

$\frac{\sqrt{10}}{x^{7}}$

Step 1

Since $\sqrt{10}$ is constant with respect to $x$ , the derivative of $\frac{\sqrt{10}}{x^{7}}$ with respect to $x$ is $\sqrt{10} \frac{d}{d x} [\frac{1}{x^{7}}]$ .

$\sqrt{10} \frac{d}{d x} [\frac{1}{x^{7}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{x^{7}}$ as ${(x^{7})}^{- 1}$ .

$\sqrt{10} \frac{d}{d x} [{(x^{7})}^{- 1}]$

Step 2.2

Multiply the exponents in ${(x^{7})}^{- 1}$ .

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Step 2.2.1

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

$\sqrt{10} \frac{d}{d x} [x^{7 \cdot - 1}]$

Step 2.2.2

Multiply $7$ by $- 1$ .

$\sqrt{10} \frac{d}{d x} [x^{- 7}]$

Step 3

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

$\sqrt{10} (- 7 x^{- 8})$

Step 4

Simplify.

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Step 4.1

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

$\sqrt{10} (- 7 \frac{1}{x^{8}})$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 7$ and $\frac{1}{x^{8}}$ .

$\sqrt{10} \frac{- 7}{x^{8}}$

Step 4.2.2

Move the negative in front of the fraction.

$\sqrt{10} (- \frac{7}{x^{8}})$

Step 4.2.3

Combine $\sqrt{10}$ and $\frac{7}{x^{8}}$ .

$- \frac{\sqrt{10} \cdot 7}{x^{8}}$

Step 4.2.4

Move $7$ to the left of $\sqrt{10}$ .

$- \frac{7 \sqrt{10}}{x^{8}}$