Calculus Examples

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

Step 1

Check if the left side of the equation is the result of the derivative of the term $x y$ .

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$x \frac{d}{d x} [y] + y \frac{d}{d x} [x]$

Step 1.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x y' + y \frac{d}{d x} [x]$

Step 1.3

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

$x y' + y \cdot 1$

Step 1.4

Substitute $\frac{d y}{d x}$ for $y'$ .

$x \frac{d y}{d x} + y \cdot 1$

Step 1.5

Reorder $y$ and $1$ .

$x \frac{d y}{d x} + 1 \cdot y$

Step 1.6

Multiply $y$ by $1$ .

$x \frac{d y}{d x} + y$

Step 2

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x y] = \frac{1}{x^{2}}$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int \frac{1}{x^{2}} d x$

Step 4

Integrate the left side.

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

Step 5

Integrate the right side.

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Step 5.1

Apply basic rules of exponents.

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Step 5.1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$x y = \int {(x^{2})}^{- 1} d x$

Step 5.1.2

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

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Step 5.1.2.1

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

$x y = \int x^{2 \cdot - 1} d x$

Step 5.1.2.2

Multiply $2$ by $- 1$ .

$x y = \int x^{- 2} d x$

Step 5.2

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$x y = - x^{- 1} + C$

Step 5.3

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

$x y = - \frac{1}{x} + C$

Step 6

Divide each term in $x y = - \frac{1}{x} + C$ by $x$ and simplify.

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Step 6.1

Divide each term in $x y = - \frac{1}{x} + C$ by $x$ .

$\frac{x y}{x} = \frac{- \frac{1}{x}}{x} + \frac{C}{x}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $x$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x y}{x} = \frac{- \frac{1}{x}}{x} + \frac{C}{x}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{1}{x}}{x} + \frac{C}{x}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{1}{x} \cdot \frac{1}{x} + \frac{C}{x}$

Step 6.3.1.2

Multiply $- \frac{1}{x} \cdot \frac{1}{x}$ .

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Step 6.3.1.2.1

Multiply $\frac{1}{x}$ by $\frac{1}{x}$ .

$y = - \frac{1}{x \cdot x} + \frac{C}{x}$

Step 6.3.1.2.2

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

$y = - \frac{1}{x^{1} x} + \frac{C}{x}$

Step 6.3.1.2.3

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

$y = - \frac{1}{x^{1} x^{1}} + \frac{C}{x}$

Step 6.3.1.2.4

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

$y = - \frac{1}{x^{1 + 1}} + \frac{C}{x}$

Step 6.3.1.2.5

Add $1$ and $1$ .

$y = - \frac{1}{x^{2}} + \frac{C}{x}$