Find the Derivative - d/da (a^x)^y(a^y)^x

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

Step 1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3

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

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

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

Add $x y$ and $y x$ .

Reorder $y$ and $x$ .

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

Add $x y$ and $x y$ .

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

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 2 x y$ .

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