Derivative of $$$y z^{3}$$$ with respect to $$$y$$$
The calculator will find the derivative of $$$y z^{3}$$$ with respect to $$$y$$$, with steps shown.
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Your Input
Find $$$\frac{d}{dy} \left(y z^{3}\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{dy} \left(c f{\left(y \right)}\right) = c \frac{d}{dy} \left(f{\left(y \right)}\right)$$$ with $$$c = z^{3}$$$ and $$$f{\left(y \right)} = y$$$:
$${\color{red}\left(\frac{d}{dy} \left(y z^{3}\right)\right)} = {\color{red}\left(z^{3} \frac{d}{dy} \left(y\right)\right)}$$Apply the power rule $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dy} \left(y\right) = 1$$$:
$$z^{3} {\color{red}\left(\frac{d}{dy} \left(y\right)\right)} = z^{3} {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{dy} \left(y z^{3}\right) = z^{3}$$$.
Answer
$$$\frac{d}{dy} \left(y z^{3}\right) = z^{3}$$$A