What is the derivative of xsin(x+y+ycos(x)) ?

The derivative of xsin(x+y+ycos(x)) is sin(x+y)+xcos(x+y)(1+(dy)/(dx))+cos(x)(dy)/(dx)-sin(x)y

What is the first derivative of xsin(x+y+ycos(x)) ?

The first derivative of xsin(x+y+ycos(x)) is sin(x+y)+xcos(x+y)(1+(dy)/(dx))+cos(x)(dy)/(dx)-sin(x)y

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\frac{d}{d x} (x sin (x + y) + y cos (x))

sin (x + y) + x cos (x + y) (1 + \frac{d y}{d x}) + cos (x) \frac{d y}{d x} - sin (x) y

Solution steps

\frac{d}{d x} (x sin (x + y) + y cos (x))

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (x sin (x + y)) + \frac{d}{d x} (y cos (x))

\frac{d}{d x} (x sin (x + y)) = sin (x + y) + x cos (x + y) (1 + \frac{d y}{d x})

\frac{d}{d x} (y cos (x)) = cos (x) \frac{d y}{d x} - sin (x) y

= sin (x + y) + x cos (x + y) (1 + \frac{d y}{d x}) + cos (x) \frac{d y}{d x} - sin (x) y

What is the derivative of xsin(x+y+ycos(x)) ?
The derivative of xsin(x+y+ycos(x)) is sin(x+y)+xcos(x+y)(1+(dy)/(dx))+cos(x)(dy)/(dx)-sin(x)y
What is the first derivative of xsin(x+y+ycos(x)) ?
The first derivative of xsin(x+y+ycos(x)) is sin(x+y)+xcos(x+y)(1+(dy)/(dx))+cos(x)(dy)/(dx)-sin(x)y