Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $z$ with respect to $x$ is $z'$ .

$z'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

By the Sum Rule, the derivative of $x^{y} + \ln (x y)$ with respect to $x$ is $\frac{d}{d x} [x^{y}] + \frac{d}{d x} [\ln (x y)]$ .

$\frac{d}{d x} [x^{y}] + \frac{d}{d x} [\ln (x y)]$

Step 3.1.2

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

$y x^{y - 1} + \frac{d}{d x} [\ln (x y)]$

Step 3.2

Evaluate $\frac{d}{d x} [\ln (x y)]$ .

Tap for more steps...

Step 3.2.1

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 $g (x) = x y$ .

Tap for more steps...

Step 3.2.1.1

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

$y x^{y - 1} + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [x y]$

Step 3.2.1.2

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

$y x^{y - 1} + \frac{1}{u} \frac{d}{d x} [x y]$

Step 3.2.1.3

Replace all occurrences of $u$ with $x y$ .

$y x^{y - 1} + \frac{1}{x y} \frac{d}{d x} [x y]$

Step 3.2.2

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

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

Step 3.2.3

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

$y x^{y - 1} + \frac{1}{x y} (y \cdot 1)$

Step 3.2.4

Multiply $y$ by $1$ .

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

Step 3.2.5

Combine $\frac{1}{x y}$ and $y$ .

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

Step 3.2.6

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.2.6.1

Cancel the common factor.

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

Step 3.2.6.2

Rewrite the expression.

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

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Combine terms.

Tap for more steps...

Step 3.3.1.1

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

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

Step 3.3.1.2

Combine the numerators over the common denominator.

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

Step 3.3.1.3

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

$\frac{y (x^{1} x^{y - 1}) + 1}{x}$

Step 3.3.1.4

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

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

Step 3.3.1.5

Subtract $1$ from $1$ .

$\frac{y x^{y + 0} + 1}{x}$

Step 3.3.1.6

Add $y$ and $0$ .

$\frac{y x^{y} + 1}{x}$

Step 3.3.2

Reorder terms.

$\frac{x^{y} y + 1}{x}$

Step 3.3.3

Reorder factors in $\frac{x^{y} y + 1}{x}$ .

$\frac{y x^{y} + 1}{x}$

Step 4

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

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

Step 5

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

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