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

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

Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps

Leave empty for autodetection.
Leave empty, if you don't need the derivative at a specific point.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

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


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

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

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

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

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

Return to the old variable:

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

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = y$$$ and $$$f{\left(x \right)} = x$$$:

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

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

$$y \cos{\left(x y \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = y \cos{\left(x y \right)} {\color{red}\left(1\right)}$$

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


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