Calculus Examples

$y z = \ln (x + z)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y z) = \frac{d}{d x} (\ln (x + z))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $z$ is constant with respect to $x$ , the derivative of $y z$ with respect to $x$ is $z \frac{d}{d x} [y]$ .

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

Step 2.2

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

$z y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x + z$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x + z$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + z]$

Step 3.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [x + z]$

Step 3.1.3

Replace all occurrences of $u$ with $x + z$ .

$\frac{1}{x + z} \frac{d}{d x} [x + z]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x + z$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [z]$ .

$\frac{1}{x + z} (\frac{d}{d x} [x] + \frac{d}{d x} [z])$

Step 3.2.2

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

$\frac{1}{x + z} (1 + \frac{d}{d x} [z])$

Step 3.2.3

Since $z$ is constant with respect to $x$ , the derivative of $z$ with respect to $x$ is $0$ .

$\frac{1}{x + z} (1 + 0)$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

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

Step 3.2.4.2

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

$z y' = \frac{1}{x + z}$

Step 5

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

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Step 5.1

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

$\frac{z y'}{z} = \frac{\frac{1}{x + z}}{z}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $z$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{z y'}{z} = \frac{\frac{1}{x + z}}{z}$

Step 5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{1}{x + z}}{z}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{1}{x + z} \cdot \frac{1}{z}$

Step 5.3.2

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

$y' = \frac{1}{(x + z) z}$

Step 5.3.3

Reorder factors in $\frac{1}{(x + z) z}$ .

$y' = \frac{1}{z (x + z)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{z (x + z)}$