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Derivative of $$$\sin{\left(u \right)} - \cos{\left(u \right)}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{du} \left(\sin{\left(u \right)} - \cos{\left(u \right)}\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)} - \cos{\left(u \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) - \frac{d}{du} \left(\cos{\left(u \right)}\right)\right)}$$The derivative of the sine is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} - \frac{d}{du} \left(\cos{\left(u \right)}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} - \frac{d}{du} \left(\cos{\left(u \right)}\right)$$The derivative of the cosine is $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$$\cos{\left(u \right)} - {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} = \cos{\left(u \right)} - {\color{red}\left(- \sin{\left(u \right)}\right)}$$Simplify:
$$\sin{\left(u \right)} + \cos{\left(u \right)} = \sqrt{2} \sin{\left(u + \frac{\pi}{4} \right)}$$Thus, $$$\frac{d}{du} \left(\sin{\left(u \right)} - \cos{\left(u \right)}\right) = \sqrt{2} \sin{\left(u + \frac{\pi}{4} \right)}$$$.
Answer
$$$\frac{d}{du} \left(\sin{\left(u \right)} - \cos{\left(u \right)}\right) = \sqrt{2} \sin{\left(u + \frac{\pi}{4} \right)}$$$A