Calculus Examples

$y^{x y}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite $y^{x y}$ as $e^{\ln (y^{x y})}$ .

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

Step 1.2

Expand $\ln (y^{x y})$ by moving $x y$ outside the logarithm.

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $x y \ln (y)$ .

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 2.3

Replace all occurrences of $u$ with $x y \ln (y)$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Differentiate using the Power Rule.

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $y$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

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

$e^{x y \ln (y)} (x (1 + \ln (y) \cdot 1))$

Step 6.4

Multiply $\ln (y)$ by $1$ .

$e^{x y \ln (y)} (x (1 + \ln (y)))$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$e^{x y \ln (y)} (x \cdot 1 + x \ln (y))$

Step 7.2

Apply the distributive property.

$e^{x y \ln (y)} (x \cdot 1) + e^{x y \ln (y)} (x \ln (y))$

Step 7.3

Multiply $x$ by $1$ .

$e^{x y \ln (y)} x + e^{x y \ln (y)} (x \ln (y))$

Step 7.4

Reorder terms.

$e^{x y \ln (y)} x + e^{x y \ln (y)} x \ln (y)$