Calculus Examples

y^{x y}

$y^{x y}$

Step 1

Differentiate using the Generalized Power Rule which states that

\frac{d}{d x} [f^{g}]

$\frac{d}{d x} [f^{g}]$ is

$f^{g} (f' \frac{g}{f} + g' \ln (f))$ where

$f = y$ and

$g = x y$ .

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

Step 2

Multiply

$y$ by

$y^{x y - 1}$ by adding the exponents.

Tap for more steps...

Step 2.1

Move

$y^{x y - 1}$ .

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

Step 2.2

Multiply

$y^{x y - 1}$ by

$y$ .

Tap for more steps...

Step 2.2.1

Raise

$y$ to the power of

$1$ .

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

Step 2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3

Combine the opposite terms in

$x y - 1 + 1$ .

Tap for more steps...

Step 2.3.1

Add

$- 1$ and

$1$ .

$\frac{d}{d x} [y] (x y^{x y + 0}) + \frac{d}{d x} [x y] (y^{x y} \ln (y))$

Step 2.3.2

Add

$x y$ and

$0$ .

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

Step 3

Since

$y$ is constant with respect to

$x$ , the derivative of

$y$ with respect to

$x$ is

$0$ .

$0 (x y^{x y}) + \frac{d}{d x} [x y] (y^{x y} \ln (y))$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Multiply

$x$ by

$0$ .

$0 y^{x y} + \frac{d}{d x} [x y] (y^{x y} \ln (y))$

Step 4.2

Multiply

$0$ by

$y^{x y}$ .

$0 + \frac{d}{d x} [x y] (y^{x y} \ln (y))$

Step 4.3

Add

$0$ and

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

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

Step 5

Since

$y$ is constant with respect to

$x$ , the derivative of

$x y$ with respect to

$x$ is

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

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

Step 6

Raise

$y$ to the power of

$1$ .

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

Step 7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Step 9

Multiply

$y^{x y + 1}$ by

$1$ .

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