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


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 $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:

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

Return to the old variable:

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

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