# Derivative of $e^{x y z}$ with respect to $z$

The calculator will find the derivative of $e^{x y z}$ with respect to $z$, with steps shown.

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Find $\frac{d}{dz} \left(e^{x y z}\right)$.

### Solution

The function $e^{x y z}$ is the composition $f{\left(g{\left(z \right)} \right)}$ of two functions $f{\left(u \right)} = e^{u}$ and $g{\left(z \right)} = x y z$.

Apply the chain rule $\frac{d}{dz} \left(f{\left(g{\left(z \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dz} \left(g{\left(z \right)}\right)$:

$${\color{red}\left(\frac{d}{dz} \left(e^{x y z}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dz} \left(x y z\right)\right)}$$

The derivative of the exponential is $\frac{d}{du} \left(e^{u}\right) = e^{u}$:

$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dz} \left(x y z\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dz} \left(x y z\right)$$

$$e^{{\color{red}\left(u\right)}} \frac{d}{dz} \left(x y z\right) = e^{{\color{red}\left(x y z\right)}} \frac{d}{dz} \left(x y z\right)$$

Apply the constant multiple rule $\frac{d}{dz} \left(c f{\left(z \right)}\right) = c \frac{d}{dz} \left(f{\left(z \right)}\right)$ with $c = x y$ and $f{\left(z \right)} = z$:

$$e^{x y z} {\color{red}\left(\frac{d}{dz} \left(x y z\right)\right)} = e^{x y z} {\color{red}\left(x y \frac{d}{dz} \left(z\right)\right)}$$

Apply the power rule $\frac{d}{dz} \left(z^{n}\right) = n z^{n - 1}$ with $n = 1$, in other words, $\frac{d}{dz} \left(z\right) = 1$:

$$x y e^{x y z} {\color{red}\left(\frac{d}{dz} \left(z\right)\right)} = x y e^{x y z} {\color{red}\left(1\right)}$$

Thus, $\frac{d}{dz} \left(e^{x y z}\right) = x y e^{x y z}$.

$\frac{d}{dz} \left(e^{x y z}\right) = x y e^{x y z}$A