Calculus Examples

$y = \frac{1}{x^{2}}$

Step 1

Set $y$ as a function of $x$ .

$f (x) = \frac{1}{x^{2}}$

Step 2

Find the derivative.

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Step 2.1

Apply basic rules of exponents.

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Step 2.1.1

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

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

Step 2.1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.1.2.1

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

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

Step 2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2

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

$- 2 x^{- 3}$

Step 2.3

Simplify.

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Step 2.3.1

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

$- 2 \frac{1}{x^{3}}$

Step 2.3.2

Combine terms.

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Step 2.3.2.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$\frac{- 2}{x^{3}}$

Step 2.3.2.2

Move the negative in front of the fraction.

$- \frac{2}{x^{3}}$

Step 3

Set the derivative equal to $0$ then solve the equation $- \frac{2}{x^{3}} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$2 = 0$

Step 3.2

Since $2 \neq 0$ , there are no solutions.

No solution

Step 4

There are no solution found by setting the derivative equal to $0$ , $- \frac{2}{x^{3}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5