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

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

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

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

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

Return to the old variable:

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

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

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

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