Calculus Examples

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

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.

$\frac{d}{d x} [x y] = \sqrt{x}$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int \sqrt{x} d x$

Step 4

Integrate the left side.

$x y = \int \sqrt{x} d x$

Step 5

Integrate the right side.

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 5.2

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$x y = \frac{2}{3} x^{\frac{3}{2}} + C$

Step 6

Divide each term in $x y = \frac{2}{3} x^{\frac{3}{2}} + C$ by $x$ and simplify.

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Step 6.1

Divide each term in $x y = \frac{2}{3} x^{\frac{3}{2}} + C$ by $x$ .

$\frac{x y}{x} = \frac{\frac{2}{3} x^{\frac{3}{2}}}{x} + \frac{C}{x}$

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.

$\frac{x y}{x} = \frac{\frac{2}{3} x^{\frac{3}{2}}}{x} + \frac{C}{x}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{2}{3} x^{\frac{3}{2}}}{x} + \frac{C}{x}$

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

Move $x^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$y = \frac{2}{3} x^{\frac{3}{2}} x^{- 1} + \frac{C}{x}$

Step 6.3.1.2

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 6.3.1.2.1

Move $x^{- 1}$ .

$y = \frac{2}{3} (x^{- 1} x^{\frac{3}{2}}) + \frac{C}{x}$

Step 6.3.1.2.2

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

$y = \frac{2}{3} x^{- 1 + \frac{3}{2}} + \frac{C}{x}$

Step 6.3.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{2}{3} x^{- 1 \cdot \frac{2}{2} + \frac{3}{2}} + \frac{C}{x}$

Step 6.3.1.2.4

Combine $- 1$ and $\frac{2}{2}$ .

$y = \frac{2}{3} x^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}} + \frac{C}{x}$

Step 6.3.1.2.5

Combine the numerators over the common denominator.

$y = \frac{2}{3} x^{\frac{- 1 \cdot 2 + 3}{2}} + \frac{C}{x}$

Step 6.3.1.2.6

Simplify the numerator.

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Step 6.3.1.2.6.1

Multiply $- 1$ by $2$ .

$y = \frac{2}{3} x^{\frac{- 2 + 3}{2}} + \frac{C}{x}$

Step 6.3.1.2.6.2

Add $- 2$ and $3$ .

$y = \frac{2}{3} x^{\frac{1}{2}} + \frac{C}{x}$

Step 6.3.1.3

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

$y = \frac{2 x^{\frac{1}{2}}}{3} + \frac{C}{x}$