Calculus Examples

$x \frac{d z}{d x} + z = 0$

Step 1

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

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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 $g (x) = z$ .

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

Step 1.2

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

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

Step 1.3

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

$x z' + z \cdot 1$

Step 1.4

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

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

Step 1.5

Reorder $z$ and $1$ .

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

Step 1.6

Multiply $z$ by $1$ .

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

Step 2

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

$\frac{d}{d x} [x z] = 0$

Step 3

Set up an integral on each side.

$\int \frac{d}{d x} [x z] d x = \int 0 d x$

Step 4

Integrate the left side.

$x z = \int 0 d x$

Step 5

Integrate the right side.

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Step 5.1

The integral of $0$ with respect to $x$ is $0$ .

$x z = 0 + C$

Step 5.2

Add $0$ and $C$ .

$x z = C$

Step 6

Divide each term in $x z = C$ by $x$ and simplify.

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Step 6.1

Divide each term in $x z = C$ by $x$ .

$\frac{x z}{x} = \frac{C}{x}$

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.

$\frac{x z}{x} = \frac{C}{x}$

Step 6.2.1.2

Divide $z$ by $1$ .

$z = \frac{C}{x}$