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

### Solution

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 $\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$:

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

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

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