Find the Derivative - d/dy e^(5xy)

$e^{5 x y}$

Step 1

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 $g (y) = 5 x y$ .

To apply the Chain Rule, set $u$ as $5 x y$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d y} [5 x y]$

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

$e^{u} \frac{d}{d y} [5 x y]$

Replace all occurrences of $u$ with $5 x y$ .

$e^{5 x y} \frac{d}{d y} [5 x y]$

Step 2

Differentiate.

Since $5 x$ is constant with respect to $y$ , the derivative of $5 x y$ with respect to $y$ is $5 x \frac{d}{d y} [y]$ .

$e^{5 x y} (5 x \frac{d}{d y} [y])$

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

$e^{5 x y} (5 x \cdot 1)$

Multiply $5$ by $1$ .

$e^{5 x y} (5 x)$

Step 3

Simplify.

Reorder the factors of $e^{5 x y} (5 x)$ .

$5 e^{5 x y} x$

Reorder factors in $5 e^{5 x y} x$ .

$5 x e^{5 x y}$