Derivative of $$$\cos{\left(x y \right)}$$$ with respect to $$$y$$$

The calculator will find the derivative of $$$\cos{\left(x y \right)}$$$ with respect to $$$y$$$, with steps shown.

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Find $$$\frac{d}{dy} \left(\cos{\left(x y \right)}\right)$$$.


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

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

$${\color{red}\left(\frac{d}{dy} \left(\cos{\left(x y \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dy} \left(x y\right)\right)}$$

The derivative of the cosine is $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dy} \left(x y\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dy} \left(x y\right)$$

Return to the old variable:

$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dy} \left(x y\right) = - \sin{\left({\color{red}\left(x y\right)} \right)} \frac{d}{dy} \left(x y\right)$$

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 $$$c = x$$$ and $$$f{\left(y \right)} = y$$$:

$$- \sin{\left(x y \right)} {\color{red}\left(\frac{d}{dy} \left(x y\right)\right)} = - \sin{\left(x y \right)} {\color{red}\left(x \frac{d}{dy} \left(y\right)\right)}$$

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

$$- x \sin{\left(x y \right)} {\color{red}\left(\frac{d}{dy} \left(y\right)\right)} = - x \sin{\left(x y \right)} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dy} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$.


$$$\frac{d}{dy} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$A