Basic Math Examples

{(a^{x})}^{y} {(a^{y})}^{x}

${(a^{x})}^{y} {(a^{y})}^{x}$

Step 1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{d}{d a} [a^{x y} a^{y x}]

$\frac{d}{d a} [a^{x y} a^{y x}]$

Step 3

Multiply

a^{x y}

$a^{x y}$ by

a^{y x}

$a^{y x}$ by adding the exponents.

Tap for more steps...

Step 3.1

Use the power rule

a^{m} a^{n} = a^{m + n}

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

\frac{d}{d a} [a^{x y + y x}]

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

Step 3.2

Add

x y

$x y$ and

y x

$y x$ .

Tap for more steps...

Step 3.2.1

Reorder

y

$y$ and

x

$x$ .

\frac{d}{d a} [a^{x y + x y}]

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

Step 3.2.2

Add

x y

$x y$ and

x y

$x y$ .

\frac{d}{d a} [a^{2 x y}]

$\frac{d}{d a} [a^{2 x y}]$

\frac{d}{d a} [a^{2 x y}]

$\frac{d}{d a} [a^{2 x y}]$

\frac{d}{d a} [a^{2 x y}]

$\frac{d}{d a} [a^{2 x y}]$

Step 4

Differentiate using the Power Rule which states that

\frac{d}{d a} [a^{n}]

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

n a^{n - 1}

$n a^{n - 1}$ where

n = 2 x y

$n = 2 x y$ .

2 x y a^{2 x y - 1}

$2 x y a^{2 x y - 1}$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$