Derivative of -x*sin(1/2) + 1

The calculator will find the derivative of $$$- x \sin{\left(\frac{1}{2} \right)} + 1$$$, with steps shown.

Your Input

Find $$$\frac{d}{dx} \left(- x \sin{\left(\frac{1}{2} \right)} + 1\right)$$$.

The trigonometric functions expect the argument in radians. To enter the argument in degrees, multiply it by pi/180, e.g. write 45° as 45*pi/180, or use the appropriate function adding 'd', e.g. write sin(45°) as sind(45).

Solution

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dx} \left(- x \sin{\left(\frac{1}{2} \right)} + 1\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(x \sin{\left(\frac{1}{2} \right)}\right) + \frac{d}{dx} \left(1\right)\right)}$$

The derivative of a constant is $$$0$$$:

$${\color{red}\left(\frac{d}{dx} \left(1\right)\right)} - \frac{d}{dx} \left(x \sin{\left(\frac{1}{2} \right)}\right) = {\color{red}\left(0\right)} - \frac{d}{dx} \left(x \sin{\left(\frac{1}{2} \right)}\right)$$

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 $$$c = \sin{\left(\frac{1}{2} \right)}$$$ and $$$f{\left(x \right)} = x$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(x \sin{\left(\frac{1}{2} \right)}\right)\right)} = - {\color{red}\left(\sin{\left(\frac{1}{2} \right)} \frac{d}{dx} \left(x\right)\right)}$$

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

$$- \sin{\left(\frac{1}{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - \sin{\left(\frac{1}{2} \right)} {\color{red}\left(1\right)}$$

Thus, $$$\frac{d}{dx} \left(- x \sin{\left(\frac{1}{2} \right)} + 1\right) = - \sin{\left(\frac{1}{2} \right)}$$$.

Answer

$$$\frac{d}{dx} \left(- x \sin{\left(\frac{1}{2} \right)} + 1\right) = - \sin{\left(\frac{1}{2} \right)}$$$A

Derivative of $$$- x \sin{\left(\frac{1}{2} \right)} + 1$$$

Your Input

Solution

Answer